Knight's tour
(Redirected from Knight's Tour)
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Problem: you have a standard 8x8 chessboard, empty but for a single knight on some square. Your task is to emit a series of legal knight moves that result in the knight visiting every square on the chessboard exactly once. Note that it is not a requirement that the tour be "closed"; that is, the knight need not end within a single move of its start position.
Input and output may be textual or graphical, according to the conventions of the programming environment. If textual, squares should be indicated in algebraic notation. The output should indicate the order in which the knight visits the squares, starting with the initial position. The form of the output may be a diagram of the board with the squares numbered according to visitation sequence, or a textual list of algebraic coordinates in order, or even an actual animation of the knight moving around the chessboard.
Input: starting square
Output: move sequence
- Related tasks
- A* search algorithm
- N-queens problem
- Solve a Hidato puzzle
- Solve a Holy Knight's tour
- Solve a Hopido puzzle
- Solve a Numbrix puzzle
- Solve the no connection puzzle
11l
V _kmoves = [(2, 1), (1, 2), (-1, 2), (-2, 1), (-2, -1), (-1, -2), (1, -2), (2, -1)]
F chess2index(=chess, boardsize)
‘Convert Algebraic chess notation to internal index format’
chess = chess.lowercase()
V x = chess[0].code - ‘a’.code
V y = boardsize - Int(chess[1..])
R (x, y)
F boardstring(board, boardsize)
V r = 0 .< boardsize
V lines = ‘’
L(y) r
lines ‘’= "\n"r.map(x -> (I @board[(x, @y)] {‘#2’.format(@board[(x, @y)])} E ‘ ’)).join(‘,’)
R lines
F knightmoves(board, P, boardsize)
V (Px, Py) = P
V kmoves = Set(:_kmoves.map((x, y) -> (@Px + x, @Py + y)))
kmoves = Set(Array(kmoves).filter((x, y) -> x C 0 .< @boardsize & y C 0 .< @boardsize & !@board[(x, y)]))
R kmoves
F accessibility(board, P, boardsize)
[(Int, (Int, Int))] access
V brd = copy(board)
L(pos) knightmoves(board, P, boardsize' boardsize)
brd[pos] = -1
access.append((knightmoves(brd, pos, boardsize' boardsize).len, pos))
brd[pos] = 0
R access
F knights_tour(start, boardsize, _debug = 0B)
[(Int, Int) = Int] board
L(x) 0 .< boardsize
L(y) 0 .< boardsize
board[(x, y)] = 0
V move = 1
V P = chess2index(start, boardsize)
board[P] = move
move++
I _debug
print(boardstring(board, boardsize' boardsize))
L move <= board.len
P = min(accessibility(board, P, boardsize))[1]
board[P] = move
move++
I _debug
print(boardstring(board, boardsize' boardsize))
input("\n#2 next: ".format(move))
R board
L(boardsize, start) [(5, ‘c3’), (8, ‘h8’), (10, ‘e6’)]
print(‘boardsize: ’boardsize)
print(‘Start position: ’start)
V board = knights_tour(start, boardsize)
print(boardstring(board, boardsize' boardsize))
print()- Output:
boardsize: 5 Start position: c3 19,12,17, 6,21 2, 7,20,11,16 13,18, 1,22, 5 8, 3,24,15,10 25,14, 9, 4,23 boardsize: 8 Start position: h8 38,41,18, 3,22,27,16, 1 19, 4,39,42,17, 2,23,26 40,37,54,21,52,25,28,15 5,20,43,56,59,30,51,24 36,55,58,53,44,63,14,29 9, 6,45,62,57,60,31,50 46,35, 8,11,48,33,64,13 7,10,47,34,61,12,49,32 boardsize: 10 Start position: e6 29, 4,57,24,73, 6,95,10,75, 8 58,23,28, 5,94,25,74, 7,100,11 3,30,65,56,27,72,99,96, 9,76 22,59, 2,63,68,93,26,81,12,97 31,64,55,66, 1,82,71,98,77,80 54,21,60,69,62,67,92,79,88,13 49,32,53,46,83,70,87,42,91,78 20,35,48,61,52,45,84,89,14,41 33,50,37,18,47,86,39,16,43,90 36,19,34,51,38,17,44,85,40,15
360 Assembly
* Knight's tour 20/03/2017
KNIGHT CSECT
USING KNIGHT,R13 base registers
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
MVC PG(20),=CL20'Knight''s tour ..x..'
L R1,NN n
XDECO R1,XDEC edit
MVC PG+14(2),XDEC+10 n
MVC PG+17(2),XDEC+10 n
XPRNT PG,L'PG print buffer
LA R0,1 1
ST R0,X x=1
ST R0,Y y=1
SR R0,R0 0
ST R0,TOTAL total=0
LOOP EQU * do loop
L R1,X x
BCTR R1,0 -1
MH R1,NNH *n
L R0,Y y
BCTR R0,0 -1
AR R1,R0 (x-1)*n+y-1
SLA R1,1 ((x-1)*n+y-1)*2
LA R0,1 1
STH R0,BOARD(R1) board(x,y)=1
L R2,TOTAL total
LA R2,1(R2) total+1
STH R2,DISP(R1) disp(x,y)=total+1
ST R2,TOTAL total=total+1
L R1,X x
L R2,Y y
BAL R14,CHOOSEMV call choosemv(x,y)
C R0,=F'0' until(choosemv(x,y)=0)
BNE LOOP loop
LA R2,KN*KN n*n
IF C,R2,NE,TOTAL THEN if total<>n*n then
XPRNT =C'error!!',7 print error
ENDIF , endif
LA R6,1 i=1
DO WHILE=(C,R6,LE,NN) do i=1 to n
MVC PG,=CL128' ' init buffer
LA R10,PG pgi=0
LA R7,1 j=1
DO WHILE=(C,R7,LE,NN) do j=1 to n
LR R1,R6 i
BCTR R1,0 -1
MH R1,NNH *n
LR R0,R7 j
BCTR R0,0 -1
AR R1,R0 (i-1)*n+j-1
SLA R1,1 ((i-1)*n+j-1)*2
LH R2,DISP(R1) disp(i,j)
XDECO R2,XDEC edit
MVC 0(4,R10),XDEC+8 output
LA R10,4(R10) pgi+=4
LA R7,1(R7) j++
ENDDO , enddo j
XPRNT PG,L'PG print buffer
LA R6,1(R6) i++
ENDDO , enddo i
L R13,4(0,R13) restore previous savearea pointer
LM R14,R12,12(R13) restore previous context
XR R15,R15 return_code=0
BR R14 exit
*------- ---- ----------------------------------------
CHOOSEMV EQU * choosemv(xc,yc)
ST R14,SAVEACMV save return point
ST R1,XC store xc
ST R2,YC store yc
MVC MM,=F'9' m=9
L R1,XC xc
LA R1,1(R1)
L R2,YC yc
LA R2,2(R2)
BAL R14,TRYMV call trymv(xc+1,yc+2)
L R1,XC xc
LA R1,1(R1)
L R2,YC yc
SH R2,=H'2'
BAL R14,TRYMV call trymv(xc+1,yc-2)
L R1,XC xc
BCTR R1,0
L R2,YC yc
LA R2,2(R2)
BAL R14,TRYMV call trymv(xc-1,yc+2)
L R1,XC xc
BCTR R1,0
L R2,YC yc
SH R2,=H'2'
BAL R14,TRYMV call trymv(xc-1,yc-2)
L R1,XC xc
LA R1,2(R1)
L R2,YC yc
LA R2,1(R2)
BAL R14,TRYMV call trymv(xc+2,yc+1)
L R1,XC xc
LA R1,2(R1)
L R2,YC yc
BCTR R2,0
BAL R14,TRYMV call trymv(xc+2,yc-1)
L R1,XC xc
SH R1,=H'2'
L R2,YC yc
LA R2,1(R2)
BAL R14,TRYMV call trymv(xc-2,yc+1)
L R1,XC xc
SH R1,=H'2'
L R2,YC yc
BCTR R2,0
BAL R14,TRYMV call trymv(xc-2,yc-1)
L R4,MM m
IF C,R4,EQ,=F'9' THEN if m=9 then
LA R0,0 return(0)
ELSE , else
MVC X,NEWX x=newx
MVC Y,NEWY y=newy
LA R0,1 return(1)
ENDIF , endif
L R14,SAVEACMV restore return point
BR R14 return
SAVEACMV DS A return point
*------- ---- ----------------------------------------
TRYMV EQU * trymv(xt,yt)
ST R14,SAVEATMV save return point
ST R1,XT store xt
ST R2,YT store yt
SR R10,R10 n=0
BAL R14,VALIDMV
IF LTR,R0,Z,R0 THEN if validmv(xt,yt)=0 then
LA R0,0 return(0)
B RETURTMV
ENDIF , endif
L R1,XT
LA R1,1(R1) xt+1
L R2,YT
LA R2,2(R2) yt+2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+1,yt+2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
LA R1,1(R1) xt+1
L R2,YT
SH R2,=H'2' yt-2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+1,yt-2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
BCTR R1,0 xt-1
L R2,YT
LA R2,2(R2) yt+2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-1,yt+2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
BCTR R1,0 xt-1
L R2,YT
SH R2,=H'2' yt-2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-1,yt-2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
LA R1,2(R1) xt+2
L R2,YT
LA R2,1(R2) yt+1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+2,yt+1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
LA R1,2(R1) xt+2
L R2,YT
BCTR R2,0 yt-1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+2,yt-1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
SH R1,=H'2' xt-2
L R2,YT
LA R2,1(R2) yt+1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-2,yt+1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
SH R1,=H'2' xt-2
L R2,YT
BCTR R2,0 yt-1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-2,yt-1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
IF C,R10,LT,MM THEN if n<m then
ST R10,MM m=n
MVC NEWX,XT newx=xt
MVC NEWY,YT newy=yt
ENDIF , endif
RETURTMV L R14,SAVEATMV restore return point
BR R14 return
SAVEATMV DS A return point
*------- ---- ----------------------------------------
VALIDMV EQU * validmv(xv,yv)
C R1,=F'1' if xv<1 then
BL RET0
C R1,NN if xv>nn then
BH RET0
C R2,=F'1' if yv<1 then
BL RET0
C R2,NN if yv>nn then
BNH OK
RET0 SR R0,R0 return(0)
B RETURVMV
OK LR R3,R1 xv
BCTR R3,0
MH R3,NNH *n
LR R0,R2 yv
BCTR R0,0
AR R3,R0
SLA R3,1
LH R4,BOARD(R3) board(xv,yv)
IF LTR,R4,Z,R4 THEN if board(xv,yv)=0 then
LA R0,1 return(1)
ELSE , else
SR R0,R0 return(0)
ENDIF , endif
RETURVMV BR R14 return
* ---- ----------------------------------------
KN EQU 8 n compile-time
NN DC A(KN) n fullword
NNH DC AL2(KN) n halfword
BOARD DC (KN*KN)H'0' dim board(n,n) init 0
DISP DC (KN*KN)H'0' dim disp(n,n) init 0
X DS F
Y DS F
TOTAL DS F
XC DS F
YC DS F
MM DS F
NEWX DS F
NEWY DS F
XT DS F
YT DS F
XDEC DS CL12
PG DC CL128' ' buffer
YREGS
END KNIGHT- Output:
Knight's tour 8x 8 1 4 57 20 47 6 49 22 34 19 2 5 58 21 46 7 3 56 35 60 37 48 23 50 18 33 38 55 52 59 8 45 39 14 53 36 61 44 51 24 32 17 40 43 54 27 62 9 13 42 15 30 11 64 25 28 16 31 12 41 26 29 10 63
Ada
First, we specify a naive implementation the package Knights_Tour with naive backtracking. It is a bit more general than required for this task, by providing a mechanism not to visit certain coordinates. This mechanism is actually useful for the task Solve a Holy Knight's tour#Ada, which also uses the package Knights_Tour.
generic
Size: Integer;
package Knights_Tour is
subtype Index is Integer range 1 .. Size;
type Tour is array (Index, Index) of Natural;
Empty: Tour := (others => (others => 0));
function Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty) return Tour;
-- finds tour via backtracking
-- either no tour has been found, i.e., Get_Tour returns Scene
-- or the Result(X,Y)=K if and only if I,J is visited at the K-th move
-- for all X, Y, Scene(X,Y) must be either 0 or Natural'Last,
-- where Scene(X,Y)=Natural'Last means "don't visit coordiates (X,Y)!"
function Count_Moves(Board: Tour) return Natural;
-- counts the number of possible moves, i.e., the number of 0's on the board
procedure Tour_IO(The_Tour: Tour; Width: Natural := 4);
-- writes The_Tour to the output using Ada.Text_IO;
end Knights_Tour;
Here is the implementation:
with Ada.Text_IO, Ada.Integer_Text_IO;
package body Knights_Tour is
type Pair is array(1..2) of Integer;
type Pair_Array is array (Positive range <>) of Pair;
Pairs: constant Pair_Array (1..8)
:= ((-2,1),(-1,2),(1,2),(2,1),(2,-1),(1,-2),(-1,-2),(-2,-1));
-- places for the night to go (relative to the current position)
function Count_Moves(Board: Tour) return Natural is
N: Natural := 0;
begin
for I in Index loop
for J in Index loop
if Board(I,J) < Natural'Last then
N := N + 1;
end if;
end loop;
end loop;
return N;
end Count_Moves;
function Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty)
return Tour is
Done: Boolean;
Move_Count: Natural := Count_Moves(Scene);
Visited: Tour;
-- Visited(I, J) = 0: not yet visited
-- Visited(I, J) = K: visited at the k-th move
-- Visited(I, J) = Integer'Last: never visit
procedure Visit(X, Y: Index; Move_Number: Positive; Found: out Boolean) is
XX, YY: Integer;
begin
Found := False;
Visited(X, Y) := Move_Number;
if Move_Number = Move_Count then
Found := True;
else
for P in Pairs'Range loop
XX := X + Pairs(P)(1);
YY := Y + Pairs(P)(2);
if (XX in Index) and then (YY in Index)
and then Visited(XX, YY) = 0 then
Visit(XX, YY, Move_Number+1, Found); -- recursion
if Found then
return; -- no need to search further
end if;
end if;
end loop;
Visited(X, Y) := 0; -- undo previous mark
end if;
end Visit;
begin
Visited := Scene;
Visit(Start_X, Start_Y, 1, Done);
if not Done then
Visited := Scene;
end if;
return Visited;
end Get_Tour;
procedure Tour_IO(The_Tour: Tour; Width: Natural := 4) is
begin
for I in Index loop
for J in Index loop
if The_Tour(I, J) < Integer'Last then
Ada.Integer_Text_IO.Put(The_Tour(I, J), Width);
else
for W in 1 .. Width-1 loop
Ada.Text_IO.Put(" ");
end loop;
Ada.Text_IO.Put("-"); -- deliberately not visited
end if;
end loop;
Ada.Text_IO.New_Line;
end loop;
end Tour_IO;
end Knights_Tour;
Here is the main program:
with Knights_Tour, Ada.Command_Line;
procedure Test_Knight is
Size: Positive := Positive'Value(Ada.Command_Line.Argument(1));
package KT is new Knights_Tour(Size => Size);
begin
KT.Tour_IO(KT.Get_Tour(1, 1));
end Test_Knight;
For small sizes, this already works well (< 1 sec for size 8). Sample output:
>./test_knight 8 1 38 55 34 3 36 19 22 54 47 2 37 20 23 4 17 39 56 33 46 35 18 21 10 48 53 40 57 24 11 16 5 59 32 45 52 41 26 9 12 44 49 58 25 62 15 6 27 31 60 51 42 29 8 13 64 50 43 30 61 14 63 28 7
For larger sizes we'll use Warnsdorff's heuristic (without any thoughtful tie breaking). We enhance the specification adding a function Warnsdorff_Get_Tour. This enhancement of the package Knights_Tour will also be used for the task Solve a Holy Knight's tour#Ada. The specification of Warnsdorff_Get_Tour is the following.
function Warnsdorff_Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty)
return Tour;
-- uses Warnsdorff heurisitic to find a tour faster
-- same interface as Get_Tour
Its implementation is as follows.
function Warnsdorff_Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty)
return Tour is
Done: Boolean;
Visited: Tour; -- see comments from Get_Tour above
Move_Count: Natural := Count_Moves(Scene);
function Neighbors(X, Y: Index) return Natural is
Result: Natural := 0;
begin
for P in Pairs'Range loop
if X+Pairs(P)(1) in Index and then Y+Pairs(P)(2) in Index and then
Visited(X+Pairs(P)(1), Y+Pairs(P)(2)) = 0 then
Result := Result + 1;
end if;
end loop;
return Result;
end Neighbors;
procedure Sort(Options: in out Pair_Array) is
N_Bors: array(Options'Range) of Natural;
K: Positive range Options'Range;
N: Natural;
P: Pair;
begin
for Opt in Options'Range loop
N_Bors(Opt) := Neighbors(Options(Opt)(1), Options(Opt)(2));
end loop;
for Opt in Options'Range loop
K := Opt;
for Alternative in Opt+1 .. Options'Last loop
if N_Bors(Alternative) < N_Bors(Opt) then
K := Alternative;
end if;
end loop;
N := N_Bors(Opt);
N_Bors(Opt) := N_Bors(K);
N_Bors(K) := N;
P := Options(Opt);
Options(Opt) := Options(K);
Options(K) := P;
end loop;
end Sort;
procedure Visit(X, Y: Index; Move: Positive; Found: out Boolean) is
Next_Count: Natural range 0 .. 8 := 0;
Next_Steps: Pair_Array(1 .. 8);
XX, YY: Integer;
begin
Found := False;
Visited(X, Y) := Move;
if Move = Move_Count then
Found := True;
else
-- consider all possible places to go
for P in Pairs'Range loop
XX := X + Pairs(P)(1);
YY := Y + Pairs(P)(2);
if (XX in Index) and then (YY in Index)
and then Visited(XX, YY) = 0 then
Next_Count := Next_Count+1;
Next_Steps(Next_Count) := (XX, YY);
end if;
end loop;
Sort(Next_Steps(1 .. Next_Count));
for N in 1 .. Next_Count loop
Visit(Next_Steps(N)(1), Next_Steps(N)(2), Move+1, Found);
if Found then
return; -- no need to search further
end if;
end loop;
-- if we didn't return above, we have to undo our move
Visited(X, Y) := 0;
end if;
end Visit;
begin
Visited := Scene;
Visit(Start_X, Start_Y, 1, Done);
if not Done then
Visited := Scene;
end if;
return Visited;
end Warnsdorff_Get_Tour;
The modification for the main program is trivial:
with Knights_Tour, Ada.Command_Line;
procedure Test_Fast is
Size: Positive := Positive'Value(Ada.Command_Line.Argument(1));
package KT is new Knights_Tour(Size => Size);
begin
KT.Tour_IO(KT.Warnsdorff_Get_Tour(1, 1));
end Test_Fast;
This works still well for somewhat larger sizes:
>./test_fast 24 1 108 45 52 3 112 57 60 5 62 131 144 7 64 147 170 9 66 187 192 11 68 71 190 46 51 2 111 56 53 4 113 130 59 6 63 146 169 8 65 186 215 10 67 188 191 12 69 107 44 109 54 123 114 129 58 61 132 145 168 143 148 185 214 171 198 225 216 193 70 189 72 50 47 122 115 110 55 140 133 128 167 142 149 184 213 172 199 226 255 246 197 224 217 194 13 43 106 49 124 139 134 127 166 141 150 183 212 173 200 227 254 247 242 223 256 245 196 73 218 48 121 116 135 126 165 138 151 182 211 174 201 228 253 248 241 290 263 304 243 222 257 14 195 105 42 125 164 137 152 181 210 175 202 229 252 249 240 289 264 329 308 291 262 303 244 219 74 120 117 136 153 180 163 176 203 230 267 250 239 288 265 328 309 334 345 330 305 292 221 258 15 41 104 119 160 177 204 231 268 209 238 287 266 251 310 335 344 357 332 307 346 261 302 75 220 118 159 154 205 162 179 208 237 286 269 324 311 336 327 438 333 418 347 356 331 306 293 16 259 103 40 161 178 207 232 285 270 323 312 337 326 483 416 343 422 437 358 419 298 349 260 301 76 158 155 206 233 284 271 236 313 338 325 482 415 342 439 484 417 420 423 348 355 360 299 294 17 39 102 157 272 235 314 339 322 481 414 341 492 497 514 421 440 485 436 359 424 297 350 77 300 156 273 234 315 276 283 478 413 340 493 480 513 530 491 498 515 452 441 454 435 354 361 18 295 101 38 275 282 397 412 321 494 479 512 557 496 543 534 529 490 499 486 451 442 425 296 351 78 274 279 316 277 320 477 410 511 570 495 554 535 556 531 542 533 516 453 444 455 434 353 362 19 37 100 281 398 411 396 575 476 567 558 561 544 553 536 521 528 489 500 487 450 443 426 79 352 280 317 278 319 402 409 510 569 560 571 566 555 550 541 532 537 522 517 460 445 456 433 20 363 99 36 389 378 399 576 395 574 475 568 559 562 545 552 525 520 527 488 501 462 449 364 427 80 94 379 318 401 388 403 408 509 572 565 474 551 540 549 538 523 518 461 446 459 432 457 366 21 35 98 93 390 377 400 573 394 375 508 563 546 373 524 519 526 371 502 463 466 365 448 81 428 380 95 382 385 404 387 376 407 564 473 374 507 548 539 372 503 464 467 370 447 458 431 22 367 383 34 97 92 391 32 405 90 393 30 547 88 471 28 505 86 469 26 465 84 369 24 429 82 96 381 384 33 386 91 392 31 406 89 472 29 506 87 470 27 504 85 468 25 430 83 368 23
ALGOL 68
# Non-recursive Knight's Tour with Warnsdorff's algorithm #
# If there are multiple choices, backtrack if the first choice doesn't #
# find a solution #
# the size of the board #
INT board size = 8;
# the vertical position changes of the moves #
# nne, nee, see, sse, ssw, sww, nww, nnw #
[]INT offset v = ( -2, -1, 1, 2, 2, 1, -1, -2 );
# the horizontal position changes of the moves #
# nne, nee, see, sse, ssw, sww, nww, nnw #
[]INT offset h = ( 1, 2, 2, 1, -1, -2, -2, -1 );
INT lowest move = LWB offset v;
INT highest move = UPB offset v;
MODE SQUARE = STRUCT( INT move # the number of the move that caused #
# the knight to reach this square #
, INT direction # the direction of the move that #
# brought the knight here - one of #
# nne, nee, see, sse, ssw, sww, nww #
# or nnw - used for backtracking #
# zero for the first move #
);
# the board #
[ board size, board size ]SQUARE board;
# initialises the board so there are no used squares #
PROC initialise board = VOID:
FOR row FROM 1 LWB board TO 1 UPB board
DO
FOR col FROM 2 LWB board TO 2 UPB board
DO
board[ row, col ] := ( 0, 0 )
OD
OD; # initialise board #
INT iterations := 0;
INT backtracks := 0;
# prints the board #
PROC print tour = VOID:
BEGIN
print( ( " a b c d e f g h", newline ) );
print( ( " +--------------------------------", newline ) );
FOR row FROM 1 UPB board BY -1 TO 1 LWB board
DO
print( ( whole( row, -3 ) ) );
print( ( "|" ) );
FOR col FROM 2 LWB board TO 2 UPB board
DO
print( ( " " ) );
print( ( whole( move OF board[ row, col ], -3 ) ) )
OD;
print( ( newline ) )
OD
END; # print tour #
# determines whether a move to the specified row and column is possible #
PROC can move to = ( INT row, INT col )BOOL:
IF row > 1 UPB board
OR row < 1 LWB board
OR col > 2 UPB board
OR col < 2 LWB board
THEN
# the position is not on the board #
FALSE
ELSE
# the move is legal, check the square is unoccupied #
move OF board[ row, col ] = 0
FI;
# used to hold counts of the number of moves that could be made in each #
# direction from the current square #
[ lowest move : highest move ]INT possible move count;
# sets the elements of possible move count to the number of moves that #
# could be made in each direction from the specified row and col #
PROC count moves in each direction from = ( INT row, INT col )VOID:
FOR move direction FROM lowest move TO highest move
DO
INT new row = row + offset v[ move direction ];
INT new col = col + offset h[ move direction ];
IF NOT can move to( new row, new col )
THEN
# can't move to this square #
possible move count[ move direction ] := -1
ELSE
# a move in this direction is possible #
# - count the number of moves that could be made from it #
possible move count[ move direction ] := 0;
FOR subsequent move FROM lowest move TO highest move
DO
IF can move to( new row + offset v[ subsequent move ]
, new col + offset h[ subsequent move ]
)
THEN
# have a possible subsequent move #
possible move count[ move direction ] +:= 1
FI
OD
FI
OD;
# update the board to the first knight's tour found starting from #
# "start row" and "start col". #
# return TRUE if one was found, FALSE otherwise #
PROC find tour = ( INT start row, INT start col )BOOL:
BEGIN
initialise board;
BOOL result := TRUE;
INT move number := 1;
INT row := start row;
INT col := start col;
# the tour will be complete when we have made as many moves #
# as there squares on the board #
INT final move = ( ( ( 1 UPB board ) + 1 ) - 1 LWB board )
* ( ( ( 2 UPB board ) + 1 ) - 2 LWB board )
;
# the first move is to place the knight on the starting square #
board[ row, col ] := ( move number, lowest move - 1 );
# start off with an unknown direction for the best move #
INT best direction := lowest move - 1;
# attempt to find a sequence of moves that will reach each square once #
WHILE
move number < final move AND result
DO
iterations +:= 1;
# count the number of moves possible from each possible move #
# from this square #
count moves in each direction from( row, col );
# find the direction with the lowest number of subsequent moves #
IF best direction < lowest move
THEN
# must find the best direction to move in #
INT lowest move count := highest move + 1;
FOR move direction FROM lowest move TO highest move
DO
IF possible move count[ move direction ] >= 0
AND possible move count[ move direction ] < lowest move count
THEN
# have a move with fewer possible subsequent moves #
best direction := move direction;
lowest move count := possible move count[ move direction ]
FI
OD
ELSE
# following a backtrack - find an alternative with the same #
# lowest number of possible moves - if there are any #
# if there aren't, we will backtrack again #
INT lowest move count := possible move count[ best direction ];
WHILE
best direction +:= 1;
IF best direction > highest move
THEN
# no more possible moves with the lowest number of #
# subsequent moves #
FALSE
ELSE
# keep looking if the number of moves from this square #
# isn't the lowest #
possible move count[ best direction ] /= lowest move count
FI
DO
SKIP
OD
FI;
IF best direction <= highest move
AND best direction >= lowest move
THEN
# we found a best possible move #
INT new row = row + offset v[ best direction ];
INT new col = col + offset h[ best direction ];
row := new row;
col := new col;
move number +:= 1;
board[ row, col ] := ( move number, best direction );
best direction := lowest move - 1
ELSE
# no more moves from this position - backtrack #
IF move number = 1
THEN
# at the starting position - no solution #
result := FALSE
ELSE
# not at the starting position - undo the latest move #
backtracks +:= 1;
move number -:= 1;
INT curr row := row;
INT curr col := col;
best direction := direction OF board[ curr row, curr col ];
row -:= offset v[ best direction ];
col -:= offset h[ best direction ];
# reset the square we just backtracked from #
board[ curr row, curr col ] := ( 0, 0 )
FI
FI
OD;
result
END; # find tour #
BEGIN
# get the starting position #
CHAR row;
CHAR col;
WHILE
print( ( "Enter starting row(1-8) and col(a-h): " ) );
read ( ( row, col, newline ) );
row < "1" OR row > "8" OR col < "a" OR col > "h"
DO
SKIP
OD;
# calculate the tour from that position, if possible #
IF find tour( ABS row - ABS "0", ( ABS col - ABS "a" ) + 1 )
THEN
# found a solution #
print tour
ELSE
# couldn't find a solution #
print( ( "Solution not found - iterations: ", iterations
, ", backtracks: ", backtracks
, newline
)
)
FI
END- Output:
Enter starting row(1-8) and col(a-h): 5d
a b c d e f g h
+--------------------------------
8| 51 18 53 20 41 44 3 6
7| 54 21 50 45 2 5 40 43
6| 17 52 19 58 49 42 7 4
5| 22 55 64 1 46 57 48 39
4| 33 16 23 56 59 38 29 8
3| 24 13 34 63 30 47 60 37
2| 15 32 11 26 35 62 9 28
1| 12 25 14 31 10 27 36 61
ATS
(*
Find Knight’s Tours.
Using Warnsdorff’s heuristic, find multiple solutions.
Optionally accept only closed tours.
Compile with:
patscc -O3 -DATS_MEMALLOC_GCBDW -o knights_tour knights_tour.dats -lgc
Usage: ./knights_tour [START_POSITION [MAX_TOURS [closed]]]
Examples:
./knights_tour (prints one tour starting from a1)
./knights_tour c5
./knights_tour c5 2000
./knights_tour c5 2000 closed
*)
#define ATS_DYNLOADFLAG 0 (* No initialization is needed. *)
#include "share/atspre_define.hats"
#include "share/atspre_staload.hats"
#define EMPTY_SQUARE ~1
macdef nil_move = @(~1, ~1)
fn
int_right_justified
{i : int}
{n : int | 0 <= n; n < 100}
(i : int i,
n : int n) :
string =
let
var buffer : @[char][100] = @[char][100] ('\0')
val _ = $extfcall (int, "snprintf", buffer, 100, "%*i", n, i)
in
strnptr2string (string1_copy ($UNSAFE.cast{string n} buffer))
end
typedef move_t (i : int,
j : int) =
@(int i, int j)
typedef move_t =
[i, j : int]
move_t (i, j)
fn
move_t_is_nil (move : move_t) :<>
bool =
let
val @(i, j) = move
val @(i_nil, j_nil) = nil_move
in
(i = i_nil && j = j_nil)
end
fn
move_t_fprint (f : FILEref,
move : move_t) :
void =
let
val @(i, j) = move
val letter = char2i 'a' + j - 1
val digit = char2i '0' + i
in
fileref_putc (f, letter);
fileref_putc (f, digit);
end
vtypedef chessboard_vt (t : t@ype,
n_ranks : int,
n_files : int,
p : addr) =
@{
pf_board = @[t][n_ranks * n_files] @ p |
n_ranks = uint n_ranks,
n_files = uint n_files,
n_squares = uint (n_ranks * n_files),
p_board = ptr p
}
vtypedef chessboard_vt (t : t@ype,
n_ranks : int,
n_files : int) =
[p : addr]
chessboard_vt (t, n_ranks, n_files, p)
vtypedef chessboard_vt (t : t@ype) =
[n_ranks, n_files : int]
chessboard_vt (t, n_ranks, n_files)
fn {t : t@ype}
chessboard_vt_make
{n_ranks, n_files : pos}
(n_ranks : uint n_ranks,
n_files : uint n_files,
fill : t) :
chessboard_vt (t, n_ranks, n_files) =
let
val size = u2sz (n_ranks * n_files)
val @(pf, pfgc | p) = array_ptr_alloc<t> (size)
val _ = array_initize_elt<t> (!p, size, fill)
prval _ = mfree_gc_v_elim pfgc (* Let the memory leak. *)
in
@{
pf_board = pf |
n_ranks = n_ranks,
n_files = n_files,
n_squares = n_ranks * n_files,
p_board = p
}
end
fn {t : t@ype}
chessboard_vt_get
{n_ranks, n_files : pos}
{i, j : int}
(chessboard : !chessboard_vt (t, n_ranks, n_files),
i : int i,
j : int j) :
t =
let
val index = (i - 1) + (u2i (chessboard.n_ranks) * (j - 1))
val _ = assertloc (0 <= index)
val _ = assertloc (index < u2i (chessboard.n_squares))
in
array_get_at (!(chessboard.p_board), index)
end
fn {t : t@ype}
chessboard_vt_set
{n_ranks, n_files : pos}
{i, j : int}
(chessboard : !chessboard_vt (t, n_ranks, n_files),
i : int i,
j : int j,
value : t) :
void =
let
val index = (i - 1) + (u2i (chessboard.n_ranks) * (j - 1))
val _ = assertloc (0 <= index)
val _ = assertloc (index < u2i (chessboard.n_squares))
in
array_set_at (!(chessboard.p_board), index, value)
end
extern fn {t : t@ype}
find_nth_position$equal (x : t,
y : t) :
bool
fn {t : t@ype}
find_nth_position
{n_ranks, n_files : pos}
(chessboard : !chessboard_vt (t, n_ranks, n_files),
n : t) :
[i, j : int]
move_t (i, j) =
let
val n_ranks = chessboard.n_ranks
val n_files = chessboard.n_files
fun
outer_loop {i : pos | i <= n_ranks + 1} .<n_ranks + 1 - i>.
(chessboard : !chessboard_vt (t, n_ranks, n_files),
i : int i) :
[i, j : int]
move_t (i, j) =
let
fun
inner_loop {j : pos | j <= n_files + 1} .<n_files + 1 - j>.
(chessboard : !chessboard_vt (t, n_ranks, n_files),
j : int j) :
[j : int]
int j =
if u2i n_files < j then
j
else
let
val v = chessboard_vt_get<t> (chessboard, i, j)
in
if find_nth_position$equal<t> (n, v) then
j
else
inner_loop (chessboard, succ j)
end
in
if u2i n_ranks < i then
nil_move
else
let
val j = inner_loop (chessboard, 1)
in
if j <= u2i n_files then
@(i, j)
else
outer_loop (chessboard, succ i)
end
end
in
outer_loop (chessboard, 1)
end
implement
find_nth_position$equal<int> (x, y) =
x = y
fn
knights_tour_is_closed
{n_ranks, n_files : pos}
(chessboard : !chessboard_vt (int, n_ranks, n_files)) :
bool =
let
val n_squares = chessboard.n_squares
val @(i1, j1) = find_nth_position<int> (chessboard, 1)
val @(i2, j2) = find_nth_position<int> (chessboard, u2i n_squares)
val i_diff = abs (i1 - i2)
val j_diff = abs (j1 - j2)
in
(i_diff = 1 && j_diff = 2) || (i_diff = 2 && j_diff = 1)
end
fn
knights_tour_board_fprint
{n_ranks, n_files : pos}
(f : FILEref,
chessboard : !chessboard_vt (int, n_ranks, n_files)) :
void =
{
val n_ranks = chessboard.n_ranks
val n_files = chessboard.n_files
fun
outer_loop {i : int | 0 <= i; i <= n_ranks} .<i>.
(chessboard : !chessboard_vt (int, n_ranks, n_files),
i : int i) :
void =
if 0 < i then
{
val _ = fileref_puts (f, " ")
val _ =
let
var j : [j : int] int j
in
for (j := 1; j <= u2i n_files; j := succ j)
fileref_puts (f, "+----")
end
val _ = fileref_puts (f, "+\n")
val _ = fileref_puts (f, int_right_justified (i, 2))
val _ = fileref_puts (f, " ")
fun
inner_loop {j : int | 1 <= j; j <= n_files + 1}
(chessboard : !chessboard_vt (int, n_ranks,
n_files),
j : int j) :
void =
if j <= u2i n_files then
{
val v = chessboard_vt_get<int> (chessboard, i, j)
val v = g1ofg0 v
val _ = fileref_puts (f, " | ")
val _ =
if v = EMPTY_SQUARE then
fileref_puts (f, " ")
else
fileref_puts (f, int_right_justified (g1ofg0 v, 2))
val _ = inner_loop (chessboard, succ j)
}
val _ = inner_loop (chessboard, 1)
val _ = fileref_puts (f, " |\n")
val _ = outer_loop (chessboard, pred i)
}
val _ = outer_loop (chessboard, u2i n_ranks)
val _ = fileref_puts (f, " ")
val _ =
let
var j : [j : int] int j
in
for (j := 1; j <= u2i n_files; j := succ j)
fileref_puts (f, "+----")
end
val _ = fileref_puts (f, "+\n")
val _ = fileref_puts (f, " ")
val _ =
let
var j : [j : int] int j
in
for (j := 1; j <= u2i n_files; j := succ j)
let
val letter = char2i 'a' + j - 1
in
fileref_puts (f, " ");
fileref_putc (f, letter)
end
end
}
fn
knights_tour_moves_fprint
{n_ranks, n_files : pos}
(f : FILEref,
chessboard : !chessboard_vt (int, n_ranks, n_files)) :
void =
{
prval _ = mul_pos_pos_pos (mul_make {n_ranks, n_files} ())
val n_ranks = chessboard.n_ranks
val n_files = chessboard.n_files
val n_squares = chessboard.n_squares
val @(pf, pfgc | p_positions) =
array_ptr_alloc<move_t> (u2sz n_squares)
val _ = array_initize_elt<move_t> (!p_positions, u2sz n_squares,
nil_move)
macdef positions = !p_positions
fun
loop {k : int | 0 <= k; k <= n_ranks * n_files}
.<n_ranks * n_files - k>.
(positions : &(@[move_t][n_ranks * n_files]),
chessboard : !chessboard_vt (int, n_ranks, n_files),
k : int k) :
void =
if k < u2i n_squares then
{
val i = u2i ((i2u k) mod n_ranks) + 1
val j = u2i ((i2u k) / n_ranks) + 1
val v = chessboard_vt_get<int> (chessboard, i, j)
val v = g1ofg0 v
val _ = assertloc (1 <= v)
val _ = assertloc (v <= u2i n_squares)
val _ = positions[v - 1] := @(i, j)
val _ = loop (positions, chessboard, succ k)
}
val _ = loop (positions, chessboard, 0)
fun
loop {k : int | 0 <= k; k < n_ranks * n_files}
.<n_ranks * n_files - k>.
(positions : &(@[move_t][n_ranks * n_files]),
k : int k) :
void =
if k < u2i (pred n_squares) then
{
val _ = move_t_fprint (f, positions[k])
val line_end = (((i2u (k + 1)) mod n_files) = 0U)
val _ =
fileref_puts (f, (if line_end then " ->\n" else " -> "))
val _ = loop (positions, succ k)
}
val _ = loop (positions, 0)
val _ = move_t_fprint (f, positions[pred n_squares])
val _ =
if knights_tour_is_closed (chessboard) then
fileref_puts (f, " -> cycle")
val _ = array_ptr_free (pf, pfgc | p_positions)
}
typedef knights_moves_t =
@(move_t, move_t, move_t, move_t,
move_t, move_t, move_t, move_t)
fn
possible_moves {n_ranks, n_files : pos}
{i, j : int}
(chessboard : !chessboard_vt (int, n_ranks, n_files),
i : int i,
j : int j) :
knights_moves_t =
let
fn
try_move {istride, jstride : int}
(chessboard : !chessboard_vt (int, n_ranks, n_files),
istride : int istride,
jstride : int jstride) :
move_t =
let
val i1 = i + istride
val j1 = j + jstride
in
if i1 < 1 then
nil_move
else if u2i (chessboard.n_ranks) < i1 then
nil_move
else if j1 < 1 then
nil_move
else if u2i (chessboard.n_files) < j1 then
nil_move
else
let
val v = chessboard_vt_get (chessboard, i1, j1) : int
in
if v <> EMPTY_SQUARE then
nil_move
else
@(i1, j1)
end
end
val move0 = try_move (chessboard, 1, 2)
val move1 = try_move (chessboard, 2, 1)
val move2 = try_move (chessboard, 1, ~2)
val move3 = try_move (chessboard, 2, ~1)
val move4 = try_move (chessboard, ~1, 2)
val move5 = try_move (chessboard, ~2, 1)
val move6 = try_move (chessboard, ~1, ~2)
val move7 = try_move (chessboard, ~2, ~1)
in
@(move0, move1, move2, move3, move4, move5, move6, move7)
end
fn
count_following_moves
{n_ranks, n_files : pos}
{i, j : int}
{n_position : int}
(chessboard : !chessboard_vt (int, n_ranks, n_files),
move : move_t (i, j),
n_position : int n_position) :
uint =
if move_t_is_nil move then
0U
else
let
fn
succ_if_move_is_not_nil
{i, j : int}
(w : uint,
move : move_t (i, j)) :<>
uint =
if move_t_is_nil move then
w
else
succ w
val @(i, j) = move
val _ = chessboard_vt_set<int> (chessboard, i, j,
succ n_position)
val following_moves = possible_moves (chessboard, i, j)
val w = 0U
val w = succ_if_move_is_not_nil (w, following_moves.0)
val w = succ_if_move_is_not_nil (w, following_moves.1)
val w = succ_if_move_is_not_nil (w, following_moves.2)
val w = succ_if_move_is_not_nil (w, following_moves.3)
val w = succ_if_move_is_not_nil (w, following_moves.4)
val w = succ_if_move_is_not_nil (w, following_moves.5)
val w = succ_if_move_is_not_nil (w, following_moves.6)
val w = succ_if_move_is_not_nil (w, following_moves.7)
val _ = chessboard_vt_set<int> (chessboard, i, j, EMPTY_SQUARE)
in
w
end
fn
pick_w (w0 : uint,
w1 : uint,
w2 : uint,
w3 : uint,
w4 : uint,
w5 : uint,
w6 : uint,
w7 : uint) :<>
uint =
let
fn
next_pick (u : uint,
v : uint) :<>
uint =
if v = 0U then
u
else if u = 0U then
v
else
min (u, v)
val w = 0U
val w = next_pick (w, w0)
val w = next_pick (w, w1)
val w = next_pick (w, w2)
val w = next_pick (w, w3)
val w = next_pick (w, w4)
val w = next_pick (w, w5)
val w = next_pick (w, w6)
val w = next_pick (w, w7)
in
w
end
fn
next_moves {n_ranks, n_files : pos}
{i, j : int}
{n_position : int}
(chessboard : !chessboard_vt (int, n_ranks, n_files),
i : int i,
j : int j,
n_position : int n_position) :
knights_moves_t =
(* Prune and sort the moves according to Warnsdorff’s heuristic,
keeping only moves that have the minimum number of legal
following moves. *)
let
val moves = possible_moves (chessboard, i, j)
val w0 = count_following_moves (chessboard, moves.0, n_position)
val w1 = count_following_moves (chessboard, moves.1, n_position)
val w2 = count_following_moves (chessboard, moves.2, n_position)
val w3 = count_following_moves (chessboard, moves.3, n_position)
val w4 = count_following_moves (chessboard, moves.4, n_position)
val w5 = count_following_moves (chessboard, moves.5, n_position)
val w6 = count_following_moves (chessboard, moves.6, n_position)
val w7 = count_following_moves (chessboard, moves.7, n_position)
val w = pick_w (w0, w1, w2, w3, w4, w5, w6, w7)
in
if w = 0U then
@(nil_move, nil_move, nil_move, nil_move,
nil_move, nil_move, nil_move, nil_move)
else
@(if w0 = w then moves.0 else nil_move,
if w1 = w then moves.1 else nil_move,
if w2 = w then moves.2 else nil_move,
if w3 = w then moves.3 else nil_move,
if w4 = w then moves.4 else nil_move,
if w5 = w then moves.5 else nil_move,
if w6 = w then moves.6 else nil_move,
if w7 = w then moves.7 else nil_move)
end
fn
make_and_fprint_tours
{n_ranks, n_files : int}
{i, j : int}
{max_tours : int}
(f : FILEref,
n_ranks : int n_ranks,
n_files : int n_files,
i : int i,
j : int j,
max_tours : int max_tours,
closed_only : bool) :
void =
{
val n_ranks = max (1, n_ranks)
val n_files = max (1, n_files)
val i = max (1, min (n_ranks, i))
val j = max (1, min (n_files, j))
val max_tours = max (1, max_tours)
val n_ranks = i2u n_ranks
val n_files = i2u n_files
val i_start = i
val j_start = j
var tours_printed : int = 0
val chessboard =
chessboard_vt_make<int> (n_ranks, n_files, g1ofg0 EMPTY_SQUARE)
fun
explore {n_ranks, n_files : pos}
{i, j : int}
{n_position : int}
(chessboard : !chessboard_vt (int, n_ranks, n_files),
i : int i,
j : int j,
n_position : int n_position,
tours_printed : &int) :
void =
if tours_printed < max_tours then
let
fn
print_board {i1, j1 : int}
(chessboard : !chessboard_vt (int, n_ranks,
n_files),
tours_printed : &int) :
void =
begin
tours_printed := succ tours_printed;
fprintln! (f, "Tour number ", tours_printed);
knights_tour_moves_fprint (f, chessboard);
fprintln! (f);
knights_tour_board_fprint (f, chessboard);
fprintln! (f);
fprintln! (f)
end
fn
satisfies_closedness
{i1, j1 : int}
(move : move_t (i1, j1)) :
bool =
if closed_only then
let
val @(i1, j1) = move
val i_diff = abs (i1 - i_start)
val j_diff = abs (j1 - j_start)
in
(i_diff = 1 && j_diff = 2)
|| (i_diff = 2 && j_diff = 1)
end
else
true
fn
try_last_move
{i1, j1 : int}
(chessboard : !chessboard_vt (int, n_ranks,
n_files),
move : move_t (i1, j1),
tours_printed : &int) :
void =
if ~move_t_is_nil move && satisfies_closedness move then
let
val @(i1, j1) = move
in
chessboard_vt_set<int> (chessboard, i1, j1,
n_position + 1);
print_board (chessboard, tours_printed);
chessboard_vt_set<int> (chessboard, i1, j1,
EMPTY_SQUARE)
end
fun
explore_inner (chessboard : !chessboard_vt (int, n_ranks,
n_files),
tours_printed : &int) :
void =
if u2i (chessboard.n_squares) - n_position = 1 then
(* Is the last move possible? If so, make it and print
the board. (Only zero or one of the moves can be
non-nil.) *)
let
val moves = possible_moves (chessboard, i, j)
in
try_last_move (chessboard, moves.0, tours_printed);
try_last_move (chessboard, moves.1, tours_printed);
try_last_move (chessboard, moves.2, tours_printed);
try_last_move (chessboard, moves.3, tours_printed);
try_last_move (chessboard, moves.4, tours_printed);
try_last_move (chessboard, moves.5, tours_printed);
try_last_move (chessboard, moves.6, tours_printed);
try_last_move (chessboard, moves.7, tours_printed)
end
else
let
val moves = next_moves (chessboard, i, j, n_position)
macdef explore_move (move) =
begin
if ~move_t_is_nil ,(move) then
explore (chessboard, (,(move)).0, (,(move)).1,
succ n_position, tours_printed)
end
in
explore_move (moves.0);
explore_move (moves.1);
explore_move (moves.2);
explore_move (moves.3);
explore_move (moves.4);
explore_move (moves.5);
explore_move (moves.6);
explore_move (moves.7)
end
in
chessboard_vt_set<int> (chessboard, i, j, n_position);
explore_inner (chessboard, tours_printed);
chessboard_vt_set<int> (chessboard, i, j, EMPTY_SQUARE)
end
val _ = explore (chessboard, i, j, 1, tours_printed)
val _ = $UNSAFE.castvwtp0{void} chessboard
}
fn
algebraic_notation_to_move (s : string) :
move_t =
let
val s = g1ofg0 s
val n = string_length s
in
if n = 2 then
let
val i = g1ofg0 (char2i (s[1]) - char2i ('0'))
val j = g1ofg0 (char2i (s[0]) - char2i ('a') + 1)
in
@(i, j)
end
else
@(1, 1)
end
implement
main0 (argc, argv) =
{
val @(i, j) =
begin
if 2 <= argc then
algebraic_notation_to_move (argv[1])
else
@(1, 1)
end : move_t
val max_tours =
begin
if 3 <= argc then
$extfcall (int, "atoi", argv[2])
else
1
end : int
val max_tours = g1ofg0 max_tours
val closed_only =
begin
if 4 <= argc then
argv[3] = "closed"
else
false
end : bool
val _ = make_and_fprint_tours (stdout_ref, 8, 8, i, j, max_tours,
closed_only)
}- Output:
$ ./knights_tour c5 2 closed
Tour number 1
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> e8 -> d6 -> b5 -> d4 -> f5 -> g7 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8 | 56 | 3 | 50 | 21 | 58 | 5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7 | 51 | 22 | 57 | 4 | 49 | 20 | 63 | 6 |
+----+----+----+----+----+----+----+----+
6 | 2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5 | 23 | 60 | 1 | 48 | 53 | 62 | 7 | 46 |
+----+----+----+----+----+----+----+----+
4 | 38 | 13 | 54 | 61 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3 | 27 | 24 | 37 | 14 | 41 | 30 | 33 | 8 |
+----+----+----+----+----+----+----+----+
2 | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1 | 25 | 28 | 11 | 40 | 15 | 32 | 9 | 34 |
+----+----+----+----+----+----+----+----+
a b c d e f g h
Tour number 2
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> b5 -> d6 -> e8 -> g7 -> f5 -> d4 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8 | 56 | 3 | 50 | 21 | 60 | 5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7 | 51 | 22 | 57 | 4 | 49 | 20 | 61 | 6 |
+----+----+----+----+----+----+----+----+
6 | 2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5 | 23 | 58 | 1 | 48 | 53 | 62 | 7 | 46 |
+----+----+----+----+----+----+----+----+
4 | 38 | 13 | 54 | 63 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3 | 27 | 24 | 37 | 14 | 41 | 30 | 33 | 8 |
+----+----+----+----+----+----+----+----+
2 | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1 | 25 | 28 | 11 | 40 | 15 | 32 | 9 | 34 |
+----+----+----+----+----+----+----+----+
a b c d e f g h
AutoHotkey
#SingleInstance, Force
#NoEnv
SetBatchLines, -1
; Uncomment if Gdip.ahk is not in your standard library
;#Include, Gdip.ahk
If !pToken := Gdip_Startup(){
MsgBox, 48, Gdiplus error!, Gdiplus failed to start. Please ensure you have Gdiplus on your system.
ExitApp
}
; I've added a simple new function here, just to ensure if anyone is having any problems then to make sure they are using the correct library version
if (Gdip_LibraryVersion() < 1.30)
{
MsgBox, 48, Version error!, Please download the latest version of the gdi+ library
ExitApp
}
OnExit, Exit
tour := "a1 b3 d2 c4 a5 b7 d8 e6 d4 b5 c7 a8 b6 c8 a7 c6 b8 a6 b4 d5 e3 d1 b2 a4 c5 d7 f8 h7 f6 g8 h6 f7 h8 g6 e7 f5 h4 g2 e1 d3 e5 g4 f2 h1 g3 f1 h2 f3 g1 h3 g5 e4 d6 e8 g7 h5 f4 e2 c1 a2 c3 b1 a3 c2 "
; Knight's tour with maximum symmetry by George Jelliss, http://www.mayhematics.com/t/8f.htm
; I know, I know, but I followed the task outline to the letter! Besides, this path is the prettiest.
; Input: starting square
InputBox, start, Knight's Tour Start, Enter Knight's starting location in algebraic notation:, , , , , , , , b3
i := InStr(tour, start)
If i=0
{
Msgbox Error, please try again.
Reload
}
; Output: move sequence
Msgbox % tour := SubStr(tour, i) . SubStr(tour, 1, i-1)
; Animation
tour .= SubStr(tour, 1, 3)
, CellSize := 30 ; pixels
, Width := Height := 9*CellSize
, TopLeftX := (A_ScreenWidth - Width) // 2
, TopLeftY := (A_ScreenHeight - Height) // 2
Gui, -Caption +E0x80000 +LastFound +AlwaysOnTop +ToolWindow +OwnDialogs
Gui, Show, NA ; show board (currently transparent)
hwnd1 := WinExist() ; required for Gdip
OnMessage(0x201, "WM_LBUTTONDOWN")
, hbm := CreateDIBSection(Width, Height)
, hdc := CreateCompatibleDC()
, obm := SelectObject(hdc, hbm)
, G := Gdip_GraphicsFromHDC(hdc)
, Gdip_SetSmoothingMode(G, 4)
Loop 1 ; remove '1' and uncomment next line to loop infinitely
{
;Gdip_GraphicsClear(G) ; uncomment to loop infinitely
cOdd := "0xFFFFCE9E" ; create brushes
, cEven := "0xFFD18B47"
, pBrushOdd := Gdip_BrushCreateSolid(cOdd)
, pBrushEven := Gdip_BrushCreateSolid(cEven)
Loop 64 ; layout board
{
Row := mod(A_Index-1,8)+1
, Col := (A_Index-1)//8+1
, Gdip_FillRectangle(G, mod(Row+Col,2) ? pBrushOdd : pBrushEven, Col * CellSize + 1, Row * CellSize + 1, CellSize - 2, CellSize - 2)
}
Gdip_DeleteBrush(pBrushOdd) ; cleanup memory
, Gdip_DeleteBrush(pBrushEven)
, UpdateLayeredWindow(hwnd1, hdc, TopLeftX, TopLeftY, Width, Height) ; update board
, pPen := Gdip_CreatePen(0x66FF0000, CellSize/10) ; create pen
, Algebraic := SubStr(tour,1,2) ; get starting coordinates
, x := (Asc(SubStr(Algebraic, 1, 1))-96+0.5)*CellSize
, y := (9.5-SubStr(Algebraic, 2, 1))*CellSize
Loop 64 ; trace path
{
Sleep, 0.5*1000
xold := x, yold := y ; a line has start and end points
, Algebraic := SubStr(tour,(A_Index)*3+1,2) ; get new coordinates
, x := (Asc(SubStr(Algebraic, 1, 1))-96+0.5)*CellSize
, y := (9.5-SubStr(Algebraic, 2, 1))*CellSize
, Gdip_DrawLine(G, pPen, xold, yold, x, y)
, UpdateLayeredWindow(hwnd1, hdc, TopLeftX, TopLeftY, Width, Height) ; update board
}
Gdip_DeletePen(pPen)
}
Return
GuiEscape:
ExitApp
Exit:
Gdip_Shutdown(pToken)
ExitApp
WM_LBUTTONDOWN()
{
If (A_Gui = 1)
PostMessage, 0xA1, 2
}
- Output:
For start at b3
b3 d2 c4 a5 b7 d8 e6 d4 b5 c7 a8 b6 c8 a7 c6 b8 a6 b4 d5 e3 d1 b2 a4 c5 d7 f8 h7 f6 g8 h6 f7 h8 g6 e7 f5 h4 g2 e1 d3 e5 g4 f2 h1 g3 f1 h2 f3 g1 h3 g5 e4 d6 e8 g7 h5 f4 e2 c1 a2 c3 b1 a3 c2 a1
... plus an animation.
AWK
# syntax: GAWK -f KNIGHTS_TOUR.AWK [-v sr=x] [-v sc=x]
#
# examples:
# GAWK -f KNIGHTS_TOUR.AWK (default)
# GAWK -f KNIGHTS_TOUR.AWK -v sr=1 -v sc=1 start at top left (default)
# GAWK -f KNIGHTS_TOUR.AWK -v sr=1 -v sc=8 start at top right
# GAWK -f KNIGHTS_TOUR.AWK -v sr=8 -v sc=8 start at bottom right
# GAWK -f KNIGHTS_TOUR.AWK -v sr=8 -v sc=1 start at bottom left
#
BEGIN {
N = 8 # board size
if (sr == "") { sr = 1 } # starting row
if (sc == "") { sc = 1 } # starting column
split("2 2 -2 -2 1 1 -1 -1",X," ")
split("1 -1 1 -1 2 -2 2 -2",Y," ")
printf("\n%dx%d board: starting row=%d col=%d\n",N,N,sr,sc)
move(sr,sc,0)
exit(1)
}
function move(x,y,m) {
if (cantMove(x,y)) {
return(0)
}
P[x,y] = ++m
if (m == N ^ 2) {
printBoard()
exit(0)
}
tryBestMove(x,y,m)
}
function cantMove(x,y) {
return( P[x,y] || x<1 || x>N || y<1 || y>N )
}
function tryBestMove(x,y,m, i) {
i = bestMove(x,y)
move(x+X[i],y+Y[i],m)
}
function bestMove(x,y, arg1,arg2,c,i,min,out) {
# Warnsdorff's rule: go to where there are fewest next moves
min = N ^ 2 + 1
for (i in X) {
arg1 = x + X[i]
arg2 = y + Y[i]
if (!cantMove(arg1,arg2)) {
c = countNext(arg1,arg2)
if (c < min) {
min = c
out = i
}
}
}
return(out)
}
function countNext(x,y, i,out) {
for (i in X) {
out += (!cantMove(x+X[i],y+Y[i]))
}
return(out)
}
function printBoard( i,j,leng) {
leng = length(N*N)
for (i=1; i<=N; i++) {
for (j=1; j<=N; j++) {
printf(" %*d",leng,P[i,j])
}
printf("\n")
}
}
output:
8x8 board: starting row=1 col=1 1 50 15 32 61 28 13 30 16 33 64 55 14 31 60 27 51 2 49 44 57 62 29 12 34 17 56 63 54 47 26 59 3 52 45 48 43 58 11 40 18 35 20 53 46 41 8 25 21 4 37 42 23 6 39 10 36 19 22 5 38 9 24 7
BASIC
ANSI BASIC
ANSI BASIC does not allow function parameters to be passed by reference, so X and Y were made global variables.
100 DECLARE EXTERNAL FUNCTION choosemove
110 !
120 RANDOMIZE
130 PUBLIC NUMERIC X, Y, TRUE, FALSE
140 LET TRUE = -1
150 LET FALSE = 0
160 !
170 SET WINDOW 1,512,1,512
180 SET AREA COLOR "black"
190 FOR x=0 TO 512-128 STEP 128
200 FOR y=0 TO 512-128 STEP 128
210 PLOT AREA:x+64,y;x+128,y;x+128,y+64;x+64,y+64
220 PLOT AREA:x,y+64;x+64,y+64;x+64,y+128;x,y+128
230 NEXT y
240 NEXT x
250 !
260 SET LINE COLOR "red"
270 SET LINE WIDTH 6
280 !
290 PUBLIC NUMERIC Board(0 TO 7,0 TO 7)
300 LET X = 0
310 LET Y = 0
320 LET Total = 0
330 DO
340 LET Board(X,Y) = TRUE
350 PLOT LINES: X*64+32,Y*64+32;
360 LET Total = Total + 1
370 LOOP UNTIL choosemove(X, Y) = FALSE
380 IF Total <> 64 THEN STOP
390 END
400 !
410 EXTERNAL FUNCTION choosemove(X1, Y1)
420 DECLARE EXTERNAL SUB trymove
430 LET M = 9
440 CALL trymove(X1+1, Y1+2, M, newx, newy)
450 CALL trymove(X1+1, Y1-2, M, newx, newy)
460 CALL trymove(X1-1, Y1+2, M, newx, newy)
470 CALL trymove(X1-1, Y1-2, M, newx, newy)
480 CALL trymove(X1+2, Y1+1, M, newx, newy)
490 CALL trymove(X1+2, Y1-1, M, newx, newy)
500 CALL trymove(X1-2, Y1+1, M, newx, newy)
510 CALL trymove(X1-2, Y1-1, M, newx, newy)
520 IF M=9 THEN
530 LET choosemove = FALSE
540 EXIT FUNCTION
550 END IF
560 LET X = newx
570 LET Y = newy
580 LET choosemove = TRUE
590 END FUNCTION
600 !
610 EXTERNAL SUB trymove(X, Y, M, newx, newy)
620 !
630 DECLARE EXTERNAL FUNCTION validmove
640 IF validmove(X,Y) = 0 THEN EXIT SUB
650 IF validmove(X+1,Y+2) <> 0 THEN LET N = N + 1
660 IF validmove(X+1,Y-2) <> 0 THEN LET N = N + 1
670 IF validmove(X-1,Y+2) <> 0 THEN LET N = N + 1
680 IF validmove(X-1,Y-2) <> 0 THEN LET N = N + 1
690 IF validmove(X+2,Y+1) <> 0 THEN LET N = N + 1
700 IF validmove(X+2,Y-1) <> 0 THEN LET N = N + 1
710 IF validmove(X-2,Y+1) <> 0 THEN LET N = N + 1
720 IF validmove(X-2,Y-1) <> 0 THEN LET N = N + 1
730 IF N>M THEN EXIT SUB
740 IF N=M AND RND<.5 THEN EXIT SUB
750 LET M = N
760 LET newx = X
770 LET newy = Y
780 END SUB
790 !
800 EXTERNAL FUNCTION validmove(X,Y)
810 LET validmove = FALSE
820 IF X<0 OR X>7 OR Y<0 OR Y>7 THEN EXIT FUNCTION
830 IF Board(X,Y)=FALSE THEN LET validmove = TRUE
840 END FUNCTION
BBC BASIC
VDU 23,22,256;256;16,16,16,128
VDU 23,23,4;0;0;0;
OFF
GCOL 4,15
FOR x% = 0 TO 512-128 STEP 128
RECTANGLE FILL x%,0,64,512
NEXT
FOR y% = 0 TO 512-128 STEP 128
RECTANGLE FILL 0,y%,512,64
NEXT
GCOL 9
DIM Board%(7,7)
X% = 0
Y% = 0
Total% = 0
REPEAT
Board%(X%,Y%) = TRUE
IF Total% DRAW X%*64+32,Y%*64+32 ELSE MOVE X%*64+32,Y%*64+32
Total% += 1
UNTIL NOT FNchoosemove(X%, Y%)
IF Total%<>64 STOP
REPEAT WAIT 1 : UNTIL FALSE
END
DEF FNchoosemove(RETURN X%, RETURN Y%)
LOCAL M%, newx%, newy%
M% = 9
PROCtrymove(X%+1, Y%+2, M%, newx%, newy%)
PROCtrymove(X%+1, Y%-2, M%, newx%, newy%)
PROCtrymove(X%-1, Y%+2, M%, newx%, newy%)
PROCtrymove(X%-1, Y%-2, M%, newx%, newy%)
PROCtrymove(X%+2, Y%+1, M%, newx%, newy%)
PROCtrymove(X%+2, Y%-1, M%, newx%, newy%)
PROCtrymove(X%-2, Y%+1, M%, newx%, newy%)
PROCtrymove(X%-2, Y%-1, M%, newx%, newy%)
IF M%=9 THEN = FALSE
X% = newx% : Y% = newy%
= TRUE
DEF PROCtrymove(X%, Y%, RETURN M%, RETURN newx%, RETURN newy%)
LOCAL N%
IF NOT FNvalidmove(X%,Y%) THEN ENDPROC
IF FNvalidmove(X%+1,Y%+2) N% += 1
IF FNvalidmove(X%+1,Y%-2) N% += 1
IF FNvalidmove(X%-1,Y%+2) N% += 1
IF FNvalidmove(X%-1,Y%-2) N% += 1
IF FNvalidmove(X%+2,Y%+1) N% += 1
IF FNvalidmove(X%+2,Y%-1) N% += 1
IF FNvalidmove(X%-2,Y%+1) N% += 1
IF FNvalidmove(X%-2,Y%-1) N% += 1
IF N%>M% THEN ENDPROC
IF N%=M% IF RND(2)=1 THEN ENDPROC
M% = N%
newx% = X% : newy% = Y%
ENDPROC
DEF FNvalidmove(X%,Y%)
IF X%<0 OR X%>7 OR Y%<0 OR Y%>7 THEN = FALSE
= NOT(Board%(X%,Y%))
Bracmat
( knightsTour
= validmoves WarnsdorffSort algebraicNotation init solve
, x y fieldsToVisit
. ~
| ( validmoves
= x y jumps moves
. !arg:(?x.?y)
& :?moves
& ( jumps
= dx dy Fs fxs fys fx fy
. !arg:(?dx.?dy)
& 1 -1:?Fs
& !Fs:?fxs
& whl
' ( !fxs:%?fx ?fxs
& !Fs:?fys
& whl
' ( !fys:%?fy ?fys
& ( (!x+!fx*!dx.!y+!fy*!dy)
: (>0:<9.>0:<9)
|
)
!moves
: ?moves
)
)
)
& jumps$(1.2)
& jumps$(2.1)
& !moves
)
& ( init
= fields x y
. :?fields
& 0:?x
& whl
' ( 1+!x:<9:?x
& 0:?y
& whl
' ( 1+!y:<9:?y
& (!x.!y) !fields:?fields
)
)
& !fields
)
& init$:?fieldsToVisit
& ( WarnsdorffSort
= sum moves elm weightedTerms
. ( weightedTerms
= pos alts fieldsToVisit moves move weight
. !arg:(%?pos ?alts.?fieldsToVisit)
& ( !fieldsToVisit:!pos
& (0.!pos)
| !fieldsToVisit:? !pos ?
& validmoves$!pos:?moves
& 0:?weight
& whl
' ( !moves:%?move ?moves
& ( !fieldsToVisit:? !move ?
& !weight+1:?weight
|
)
)
& (!weight.!pos)
| 0
)
+ weightedTerms$(!alts.!fieldsToVisit)
| 0
)
& weightedTerms$!arg:?sum
& :?moves
& whl
' ( !sum:(#.?elm)+?sum
& !moves !elm:?moves
)
& !moves
)
& ( solve
= pos alts fieldsToVisit A Z tailOfSolution
. !arg:(%?pos ?alts.?fieldsToVisit)
& ( !fieldsToVisit:?A !pos ?Z
& ( !A !Z:&
| solve
$ ( WarnsdorffSort$(validmoves$!pos.!A !Z)
. !A !Z
)
)
| solve$(!alts.!fieldsToVisit)
)
: ?tailOfSolution
& !pos !tailOfSolution
)
& ( algebraicNotation
= x y
. !arg:(?x.?y) ?arg
& str$(chr$(asc$a+!x+-1) !y " ")
algebraicNotation$!arg
|
)
& @(!arg:?x #?y)
& asc$!x+-1*asc$a+1:?x
& str
$ (algebraicNotation$(solve$((!x.!y).!fieldsToVisit)))
)
& out$(knightsTour$a1);a1 b3 a5 b7 d8 f7 h8 g6 f8 h7 g5 h3 g1 e2 c1 a2 b4 a6 b8 c6 a7 c8 e7 g8 h6 g4 h2 f1 d2 b1 a3 c2 e1 f3 h4 g2 e3 d1 b2 a4 c3 b5 d4 f5 d6 c4 e5 d3 f2 h1 g3 e4 c5 d7 b6 a8 c7 d5 f4 e6 g7 e8 f6 h5
C
For an animated version using OpenGL, see Knight's tour/C.
The following draws on console the progress of the horsie. Specify board size on commandline, or use default 8.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <unistd.h>
typedef unsigned char cell;
int dx[] = { -2, -2, -1, 1, 2, 2, 1, -1 };
int dy[] = { -1, 1, 2, 2, 1, -1, -2, -2 };
void init_board(int w, int h, cell **a, cell **b)
{
int i, j, k, x, y, p = w + 4, q = h + 4;
/* b is board; a is board with 2 rows padded at each side */
a[0] = (cell*)(a + q);
b[0] = a[0] + 2;
for (i = 1; i < q; i++) {
a[i] = a[i-1] + p;
b[i] = a[i] + 2;
}
memset(a[0], 255, p * q);
for (i = 0; i < h; i++) {
for (j = 0; j < w; j++) {
for (k = 0; k < 8; k++) {
x = j + dx[k], y = i + dy[k];
if (b[i+2][j] == 255) b[i+2][j] = 0;
b[i+2][j] += x >= 0 && x < w && y >= 0 && y < h;
}
}
}
}
#define E "\033["
int walk_board(int w, int h, int x, int y, cell **b)
{
int i, nx, ny, least;
int steps = 0;
printf(E"H"E"J"E"%d;%dH"E"32m[]"E"m", y + 1, 1 + 2 * x);
while (1) {
/* occupy cell */
b[y][x] = 255;
/* reduce all neighbors' neighbor count */
for (i = 0; i < 8; i++)
b[ y + dy[i] ][ x + dx[i] ]--;
/* find neighbor with lowest neighbor count */
least = 255;
for (i = 0; i < 8; i++) {
if (b[ y + dy[i] ][ x + dx[i] ] < least) {
nx = x + dx[i];
ny = y + dy[i];
least = b[ny][nx];
}
}
if (least > 7) {
printf(E"%dH", h + 2);
return steps == w * h - 1;
}
if (steps++) printf(E"%d;%dH[]", y + 1, 1 + 2 * x);
x = nx, y = ny;
printf(E"%d;%dH"E"31m[]"E"m", y + 1, 1 + 2 * x);
fflush(stdout);
usleep(120000);
}
}
int solve(int w, int h)
{
int x = 0, y = 0;
cell **a, **b;
a = malloc((w + 4) * (h + 4) + sizeof(cell*) * (h + 4));
b = malloc((h + 4) * sizeof(cell*));
while (1) {
init_board(w, h, a, b);
if (walk_board(w, h, x, y, b + 2)) {
printf("Success!\n");
return 1;
}
if (++x >= w) x = 0, y++;
if (y >= h) {
printf("Failed to find a solution\n");
return 0;
}
printf("Any key to try next start position");
getchar();
}
}
int main(int c, char **v)
{
int w, h;
if (c < 2 || (w = atoi(v[1])) <= 0) w = 8;
if (c < 3 || (h = atoi(v[2])) <= 0) h = w;
solve(w, h);
return 0;
}
C#
using System;
using System.Collections.Generic;
namespace prog
{
class MainClass
{
const int N = 8;
readonly static int[,] moves = { {+1,-2},{+2,-1},{+2,+1},{+1,+2},
{-1,+2},{-2,+1},{-2,-1},{-1,-2} };
struct ListMoves
{
public int x, y;
public ListMoves( int _x, int _y ) { x = _x; y = _y; }
}
public static void Main (string[] args)
{
int[,] board = new int[N,N];
board.Initialize();
int x = 0, // starting position
y = 0;
List<ListMoves> list = new List<ListMoves>(N*N);
list.Add( new ListMoves(x,y) );
do
{
if ( Move_Possible( board, x, y ) )
{
int move = board[x,y];
board[x,y]++;
x += moves[move,0];
y += moves[move,1];
list.Add( new ListMoves(x,y) );
}
else
{
if ( board[x,y] >= 8 )
{
board[x,y] = 0;
list.RemoveAt(list.Count-1);
if ( list.Count == 0 )
{
Console.WriteLine( "No solution found." );
return;
}
x = list[list.Count-1].x;
y = list[list.Count-1].y;
}
board[x,y]++;
}
}
while( list.Count < N*N );
int last_x = list[0].x,
last_y = list[0].y;
string letters = "ABCDEFGH";
for( int i=1; i<list.Count; i++ )
{
Console.WriteLine( string.Format("{0,2}: ", i) + letters[last_x] + (last_y+1) + " - " + letters[list[i].x] + (list[i].y+1) );
last_x = list[i].x;
last_y = list[i].y;
}
}
static bool Move_Possible( int[,] board, int cur_x, int cur_y )
{
if ( board[cur_x,cur_y] >= 8 )
return false;
int new_x = cur_x + moves[board[cur_x,cur_y],0],
new_y = cur_y + moves[board[cur_x,cur_y],1];
if ( new_x >= 0 && new_x < N && new_y >= 0 && new_y < N && board[new_x,new_y] == 0 )
return true;
return false;
}
}
}
C++
Uses Warnsdorff's rule and (iterative) backtracking if that fails.
#include <iostream>
#include <iomanip>
#include <array>
#include <string>
#include <tuple>
#include <algorithm>
using namespace std;
template<int N = 8>
class Board
{
public:
array<pair<int, int>, 8> moves;
array<array<int, N>, N> data;
Board()
{
moves[0] = make_pair(2, 1);
moves[1] = make_pair(1, 2);
moves[2] = make_pair(-1, 2);
moves[3] = make_pair(-2, 1);
moves[4] = make_pair(-2, -1);
moves[5] = make_pair(-1, -2);
moves[6] = make_pair(1, -2);
moves[7] = make_pair(2, -1);
}
array<int, 8> sortMoves(int x, int y) const
{
array<tuple<int, int>, 8> counts;
for(int i = 0; i < 8; ++i)
{
int dx = get<0>(moves[i]);
int dy = get<1>(moves[i]);
int c = 0;
for(int j = 0; j < 8; ++j)
{
int x2 = x + dx + get<0>(moves[j]);
int y2 = y + dy + get<1>(moves[j]);
if (x2 < 0 || x2 >= N || y2 < 0 || y2 >= N)
continue;
if(data[y2][x2] != 0)
continue;
c++;
}
counts[i] = make_tuple(c, i);
}
// Shuffle to randomly break ties
random_shuffle(counts.begin(), counts.end());
// Lexicographic sort
sort(counts.begin(), counts.end());
array<int, 8> out;
for(int i = 0; i < 8; ++i)
out[i] = get<1>(counts[i]);
return out;
}
void solve(string start)
{
for(int v = 0; v < N; ++v)
for(int u = 0; u < N; ++u)
data[v][u] = 0;
int x0 = start[0] - 'a';
int y0 = N - (start[1] - '0');
data[y0][x0] = 1;
array<tuple<int, int, int, array<int, 8>>, N*N> order;
order[0] = make_tuple(x0, y0, 0, sortMoves(x0, y0));
int n = 0;
while(n < N*N-1)
{
int x = get<0>(order[n]);
int y = get<1>(order[n]);
bool ok = false;
for(int i = get<2>(order[n]); i < 8; ++i)
{
int dx = moves[get<3>(order[n])[i]].first;
int dy = moves[get<3>(order[n])[i]].second;
if(x+dx < 0 || x+dx >= N || y+dy < 0 || y+dy >= N)
continue;
if(data[y + dy][x + dx] != 0)
continue;
get<2>(order[n]) = i + 1;
++n;
data[y+dy][x+dx] = n + 1;
order[n] = make_tuple(x+dx, y+dy, 0, sortMoves(x+dx, y+dy));
ok = true;
break;
}
if(!ok) // Failed. Backtrack.
{
data[y][x] = 0;
--n;
}
}
}
template<int N>
friend ostream& operator<<(ostream &out, const Board<N> &b);
};
template<int N>
ostream& operator<<(ostream &out, const Board<N> &b)
{
for (int v = 0; v < N; ++v)
{
for (int u = 0; u < N; ++u)
{
if (u != 0) out << ",";
out << setw(3) << b.data[v][u];
}
out << endl;
}
return out;
}
int main()
{
Board<5> b1;
b1.solve("c3");
cout << b1 << endl;
Board<8> b2;
b2.solve("b5");
cout << b2 << endl;
Board<31> b3; // Max size for <1000 squares
b3.solve("a1");
cout << b3 << endl;
return 0;
}
Output:
23, 16, 11, 6, 21 10, 5, 22, 17, 12 15, 24, 1, 20, 7 4, 9, 18, 13, 2 25, 14, 3, 8, 19 63, 20, 3, 24, 59, 36, 5, 26 2, 23, 64, 37, 4, 25, 58, 35 19, 62, 21, 50, 55, 60, 27, 6 22, 1, 54, 61, 38, 45, 34, 57 53, 18, 49, 44, 51, 56, 7, 28 12, 15, 52, 39, 46, 31, 42, 33 17, 48, 13, 10, 43, 40, 29, 8 14, 11, 16, 47, 30, 9, 32, 41 275,112, 19,116,277,604, 21,118,823,770, 23,120,961,940, 25,122,943,926, 27,124,917,898, 29,126,911,872, 31,128,197,870, 33 18,115,276,601, 20,117,772,767, 22,119,958,851, 24,121,954,941, 26,123,936,925, 28,125,912,899, 30,127,910,871, 32,129,198 111,274,113,278,605,760,603,822,771,824,769,948,957,960,939,944,953,942,927,916,929,918,897,908,913,900,873,196,875, 34,869 114, 17,600,273,602,775,766,773,768,949,850,959,852,947,952,955,932,937,930,935,924,915,920,905,894,909,882,901,868,199,130 271,110,279,606,759,610,761,776,821,764,825,816,951,956,853,938,945,934,923,928,919,896,893,914,907,904,867,874,195,876, 35 16,581,272,599,280,607,774,765,762,779,950,849,826,815,946,933,854,931,844,857,890,921,906,895,886,883,902,881,200,131,194 109,270,281,580,609,758,611,744,777,820,763,780,817,848,827,808,811,846,855,922,843,858,889,892,903,866,885,192,877, 36,201 282, 15,582,269,598,579,608,757,688,745,778,819,754,783,814,847,828,807,810,845,856,891,842,859,884,887,880,863,202,193,132 267,108,283,578,583,612,689,614,743,756,691,746,781,818,753,784,809,812,829,806,801,840,835,888,865,862,203,878,191,530, 37 14,569,268,585,284,597,576,619,690,687,742,755,692,747,782,813,752,785,802,793,830,805,860,841,836,879,864,529,204,133,190 107,266,285,570,577,584,613,686,615,620,695,684,741,732,711,748,739,794,751,786,803,800,839,834,861,528,837,188,531, 38,205 286, 13,568,265,586,575,596,591,618,685,616,655,696,693,740,733,712,749,738,795,792,831,804,799,838,833,722,527,206,189,134 263,106,287,508,571,590,587,574,621,592,639,694,683,656,731,710,715,734,787,750,737,796,791,832,721,798,207,532,187,474, 39 12,417,264,567,288,509,572,595,588,617,654,657,640,697,680,713,730,709,716,735,788,727,720,797,790,723,526,473,208,135,186 105,262,289,416,507,566,589,512,573,622,593,638,653,682,659,698,679,714,729,708,717,736,789,726,719,472,533,184,475, 40,209 290, 11,418,261,502,415,510,565,594,513,562,641,658,637,652,681,660,699,678,669,728,707,718,675,724,525,704,471,210,185,136 259,104,291,414,419,506,503,514,511,564,623,548,561,642,551,636,651,670,661,700,677,674,725,706,703,534,211,476,183,396, 41 10,331,260,493,292,501,420,495,504,515,498,563,624,549,560,643,662,635,650,671,668,701,676,673,524,705,470,395,212,137,182 103,258,293,330,413,494,505,500,455,496,547,516,485,552,625,550,559,644,663,634,649,672,667,702,535,394,477,180,397, 42,213 294, 9,332,257,492,329,456,421,490,499,458,497,546,517,484,553,626,543,558,645,664,633,648,523,666,469,536,393,220,181,138 255,102,295,328,333,412,491,438,457,454,489,440,459,486,545,518,483,554,627,542,557,646,665,632,537,478,221,398,179,214, 43 8,319,256,335,296,345,326,409,422,439,436,453,488,441,460,451,544,519,482,555,628,541,522,647,468,631,392,219,222,139,178 101,254,297,320,327,334,411,346,437,408,423,368,435,452,487,442,461,450,445,520,481,556,629,538,479,466,399,176,215, 44,165 298, 7,318,253,336,325,344,349,410,347,360,407,424,383,434,427,446,443,462,449,540,521,480,467,630,391,218,223,164,177,140 251,100,303,300,321,316,337,324,343,350,369,382,367,406,425,384,433,428,447,444,463,430,539,390,465,400,175,216,169,166, 45 6,299,252,317,304,301,322,315,348,361,342,359,370,381,366,405,426,385,432,429,448,389,464,401,174,217,224,163,150,141,168 99,250,241,302,235,248,307,338,323,314,351,362,341,358,371,380,365,404,377,386,431,402,173,388,225,160,153,170,167, 46,143 240, 5, 98,249,242,305,234,247,308,339,232,313,352,363,230,357,372,379,228,403,376,387,226,159,154,171,162,149,142,151, 82 63, 2,239, 66, 97,236,243,306,233,246,309,340,231,312,353,364,229,356,373,378,227,158,375,172,161,148,155,152, 83,144, 47 4, 67, 64, 61,238, 69, 96, 59,244, 71, 94, 57,310, 73, 92, 55,354, 75, 90, 53,374, 77, 88, 51,156, 79, 86, 49,146, 81, 84 1, 62, 3, 68, 65, 60,237, 70, 95, 58,245, 72, 93, 56,311, 74, 91, 54,355, 76, 89, 52,157, 78, 87, 50,147, 80, 85, 48,145
Common Lisp
This interactive program will ask for a starting case in algebraic notation and, also, whether a closed tour is desired. Each next move is selected according to Warnsdorff's rule; ties are broken at random.
The closed tour algorithm is quite crude: just find tours over and over until one happens to be closed by chance.
This code is quite verbose: I tried to make it easy for myself and for others to follow and understand. I'm not a Lisp expert, so I probably missed some idiomatic shortcuts I could have used to make it shorter.
For some reason, the interactive part does not work with SBCL, but it works fine with CLISP.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Solving the knight's tour. ;;;
;;; Warnsdorff's rule with random tie break. ;;;
;;; Optionally outputs a closed tour. ;;;
;;; Options from interactive prompt. ;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defparameter *side* 8)
(defun generate-chessboard (n)
(loop for i below n append
(loop for j below n collect (complex i j))))
(defparameter *chessboard*
(generate-chessboard *side*))
(defun complex->algebraic (n)
;; returns a string like "b2"
(concatenate 'string
;; 'a' is char #97: add it to the offset
(string (character (+ 97 (realpart n))))
;; indices start at 0, but algebraic starts at 1
(string (digit-char (+ 1 (imagpart n))))))
(defun algebraic->complex (string)
;; takes a string like "e4"
(let ((row (char string 0))
(col (char string 1)))
(complex (- (char-code row) 97)
(- (digit-char-p col) 1))))
(defconstant *knight-directions*
(list
(complex 1 2)
(complex 2 1)
(complex 1 -2)
(complex 2 -1)
(complex -1 2)
(complex -2 1)
(complex -1 -2)
(complex -2 -1)))
(defun find-legal-moves (moves-list)
;; 2. the move must not be on a case already visited
(remove-if (lambda (m) (member m moves-list))
;; 1. the move must be within the chessboard
(intersection
(mapcar (lambda (i) (+ (car moves-list) i)) *knight-directions*)
*chessboard*)))
;; Select between two moves by Warnsdorff's rule:
;; pick the one with the lowest index or else
;; randomly break the tie.
;; Takes a cons in the form (n . #C(x y)).
;; This will be the sorting rule for picking the next move.
(defun w-rule (a b)
(cond ((< (car a) (car b)) t)
((> (car a) (car b)) nil)
((= (car a) (car b))
(zerop (random 2)))))
;; For every legal move in a given position,
;; look forward one move and return a cons
;; in the form (n . #C(x y)) where n is
;; how many next free moves follow the first move.
(defun return-weighted-moves (moves)
(let ((candidates (find-legal-moves moves)))
(loop for mv in candidates collect
(cons
(list-length (find-legal-moves (cons mv moves)))
mv))))
;; Given a list of weighted moves (as above),
;; pick one according to the w-rule
(defun pick-among-weighted-moves (moves)
;; prune dead ends one move early
(let ((possible-moves
(remove-if (lambda(m) (zerop (car m))) moves)))
(cdar (sort possible-moves #'w-rule))))
(defun make-move (moves-list)
(let ((next-move
(if (< (list-length moves-list) (1- (list-length *chessboard*)))
(pick-among-weighted-moves (return-weighted-moves moves-list))
(car (find-legal-moves moves-list)))))
(cons next-move moves-list)))
(defun make-tour (moves-list)
;; takes a list of moves as an argument
(if (null (car moves-list)) ; last move not found: start over
(make-tour (last moves-list))
(if (= (list-length moves-list) (list-length *chessboard*))
moves-list
(make-tour (make-move moves-list)))))
(defun make-closed-tour (moves-list)
(let ((tour (make-tour moves-list)))
(if (tour-closed-p tour)
tour
(make-closed-tour moves-list))))
(defun tour-closed-p (tour)
;; takes a full tour as an argument
(let ((start (car (last tour)))
(end (car tour)))
;; is the first position a legal move, when
;; viewed from the last move?
(if (member start (find-legal-moves (list end))) ; find-legal-moves takes a list
t nil)))
(defun print-tour-linear (tour)
;; takes a tour (moves list) with the last move first
;; and prints it nicely in algebraic notation
(let ((moves (mapcar #'complex->algebraic (reverse tour))))
(format t "~{~A~^ -> ~}" moves)))
(defun tour->matrix (tour)
;; takes a tour and makes a row-by-row 2D matrix
;; from top to bottom (for further formatting & printing)
(flet ((index-tour (tour) ; 1st local function
(loop for i below (length tour)
;; starting from index 1, not 0, so add 1;
;; reverse because the last move is still in the car
collect (cons (nth i (reverse tour)) (1+ i))))
(get-row (n tour) ; 2nd local function
;; in every row, the imaginary part (vertical offset) stays the same
(remove-if-not (lambda (e) (= n (imagpart (car e)))) tour)))
(let* ((indexed-tour (index-tour tour))
(ordered-indexed-tour
;; make a list of ordered rows
(loop for i from (1- *side*) downto 0 collect
(sort (get-row i indexed-tour)
(lambda (a b) (< (realpart (car a)) (realpart (car b))))))))
;; clean up, leaving only the indices
(mapcar (lambda (e) (mapcar #'cdr e)) ordered-indexed-tour))))
(defun print-tour-matrix (tour)
(mapcar (lambda (row)
(format t "~{~3d~}~&" row)) (tour->matrix tour)))
;;; Handling options
(defstruct options
closed
start
grid)
(defparameter *opts* (make-options))
;;; Interactive part
(defun prompt()
(format t "Starting case (leave blank for random)? ")
(let ((start (string (read-line))))
(if (member start (mapcar #'complex->algebraic *chessboard*) :test #'equal)
(setf (options-start *opts*) start))
(format t "Require a closed tour (yes or default to no)? ")
(let ((closed (read-line)))
(if (or (equal closed "y") (equal closed "yes"))
(setf (options-closed *opts*) t)))))
(defun main ()
(let* ((start
(if (options-start *opts*)
(algebraic->complex (options-start *opts*))
(complex (random *side*) (random *side*))))
(closed (options-closed *opts*))
(tour
(if closed
(make-closed-tour (list start))
(make-tour (list start)))))
(fresh-line)
(if closed (princ "Closed "))
(princ "Knight's tour")
(if (options-start *opts*)
(princ ":")
(princ " (starting on a random case):"))
(fresh-line)
(print-tour-linear tour)
(princ #\newline)
(princ #\newline)
(print-tour-matrix tour)))
;;; Good to go: invocation!
(prompt)
(main)
- Output:
Starting case (leave blank for random)? a8 Require a closed tour (yes or default to no)? y Closed Knight's tour: a8 -> c7 -> e8 -> g7 -> h5 -> g3 -> h1 -> f2 -> h3 -> g1 -> e2 -> c1 -> a2 -> b4 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> e6 -> d8 -> b7 -> a5 -> b3 -> a1 -> c2 -> e1 -> g2 -> f4 -> d3 -> c5 -> a4 -> b2 -> d1 -> c3 -> b1 -> a3 -> b5 -> a7 -> c6 -> d4 -> f3 -> h4 -> g6 -> h8 -> f7 -> e5 -> g4 -> h2 -> f1 -> d2 -> e4 -> f6 -> g8 -> h6 -> f5 -> e7 -> d5 -> e3 -> c4 -> d6 -> c8 -> b6 1 16 63 22 3 18 55 46 40 23 2 17 58 47 4 19 15 64 41 62 21 54 45 56 24 39 32 59 48 57 20 5 33 14 61 42 53 30 49 44 38 25 36 31 60 43 6 9 13 34 27 52 11 8 29 50 26 37 12 35 28 51 10 7
Clojure
Using warnsdorff's rule
(defn isin? [x li]
(not= [] (filter #(= x %) li)))
(defn options [movements pmoves n]
(let [x (first (last movements)) y (second (last movements))
op (vec (map #(vector (+ x (first %)) (+ y (second %))) pmoves))
vop (filter #(and (>= (first %) 0) (>= (last %) 0)) op)
vop1 (filter #(and (< (first %) n) (< (last %) n)) vop)]
(vec (filter #(not (isin? % movements)) vop1))))
(defn next-move [movements pmoves n]
(let [op (options movements pmoves n)
sp (map #(vector % (count (options (conj movements %) pmoves n))) op)
m (apply min (map last sp))]
(first (rand-nth (filter #(= m (last %)) sp)))))
(defn jumps [n pos]
(let [movements (vector pos)
pmoves [[1 2] [1 -2] [2 1] [2 -1]
[-1 2] [-1 -2] [-2 -1] [-2 1]]]
(loop [mov movements x 1]
(if (= x (* n n))
mov
(let [np (next-move mov pmoves n)]
(recur (conj mov np) (inc x)))))))
- Output:
(jumps 5 [0 0])
[[0 0] [1 2] [0 4] [2 3] [4 4] [3 2] [4 0] [2 1] [1 3] [0 1] [2 0] [4 1] [3 3] [1 4] [0 2] [1 0] [3 1] [4 3] [2 4] [0 3] [1 1] [3 0] [4 2] [3 4] [2 2]]
(jumps 8 [0 0])
[[0 0] [2 1] [4 0] [6 1] [7 3] [6 5] [7 7] [5 6] [3 7] [1 6] [0 4] [1 2] [2 0] [0 1] [1 3] [0 5] [1 7] [2 5] [0 6] [2 7] [4 6] [6 7] [7 5] [6 3] [7 1] [5 0] [3 1] [1 0] [0 2] [1 4] [3 5] [4 7] [6 6] [7 4] [6 2] [7 0] [5 1] [7 2] [6 0] [4 1] [5 3] [3 2] [4 4] [5 2] [3 3] [5 4] [4 2] [2 3] [1 1] [3 0] [2 2] [0 3] [2 4] [4 3] [6 4] [4 5] [2 6] [0 7] [1 5] [3 4] [5 5] [7 6] [5 7] [3 6]]
(let [j (jumps 40 [0 0])] ;; are
(and (distinct? j) ;; all squares only once? and
(= (count j) (* 40 40)))) ;; 40*40 squares?
true
CoffeeScript
This algorithm finds 100,000 distinct solutions to the 8x8 problem in about 30 seconds. It precomputes knight moves up front, so it turns into a pure graph traversal problem. The program uses iteration and backtracking to find solutions.
graph_tours = (graph, max_num_solutions) ->
# graph is an array of arrays
# graph[3] = [4, 5] means nodes 4 and 5 are reachable from node 3
#
# Returns an array of tours (up to max_num_solutions in size), where
# each tour is an array of nodes visited in order, and where each
# tour visits every node in the graph exactly once.
#
complete_tours = []
visited = (false for node in graph)
dead_ends = ({} for node in graph)
tour = [0]
valid_neighbors = (i) ->
arr = []
for neighbor in graph[i]
continue if visited[neighbor]
continue if dead_ends[i][neighbor]
arr.push neighbor
arr
next_square_to_visit = (i) ->
arr = valid_neighbors i
return null if arr.length == 0
# We traverse to our neighbor who has the fewest neighbors itself.
fewest_neighbors = valid_neighbors(arr[0]).length
neighbor = arr[0]
for i in [1...arr.length]
n = valid_neighbors(arr[i]).length
if n < fewest_neighbors
fewest_neighbors = n
neighbor = arr[i]
neighbor
while tour.length > 0
current_square = tour[tour.length - 1]
visited[current_square] = true
next_square = next_square_to_visit current_square
if next_square?
tour.push next_square
if tour.length == graph.length
complete_tours.push (n for n in tour) # clone
break if complete_tours.length == max_num_solutions
# pessimistically call this a dead end
dead_ends[current_square][next_square] = true
current_square = next_square
else
# we backtrack
doomed_square = tour.pop()
dead_ends[doomed_square] = {}
visited[doomed_square] = false
complete_tours
knight_graph = (board_width) ->
# Turn the Knight's Tour into a pure graph-traversal problem
# by precomputing all the legal moves. Returns an array of arrays,
# where each element in any subarray is the index of a reachable node.
index = (i, j) ->
# index squares from 0 to n*n - 1
board_width * i + j
reachable_squares = (i, j) ->
deltas = [
[ 1, 2]
[ 1, -2]
[ 2, 1]
[ 2, -1]
[-1, 2]
[-1, -2]
[-2, 1]
[-2, -1]
]
neighbors = []
for delta in deltas
[di, dj] = delta
ii = i + di
jj = j + dj
if 0 <= ii < board_width
if 0 <= jj < board_width
neighbors.push index(ii, jj)
neighbors
graph = []
for i in [0...board_width]
for j in [0...board_width]
graph[index(i, j)] = reachable_squares i, j
graph
illustrate_knights_tour = (tour, board_width) ->
pad = (n) ->
return " _" if !n?
return " " + n if n < 10
"#{n}"
console.log "\n------"
moves = {}
for square, i in tour
moves[square] = i + 1
for i in [0...board_width]
s = ''
for j in [0...board_width]
s += " " + pad moves[i*board_width + j]
console.log s
BOARD_WIDTH = 8
MAX_NUM_SOLUTIONS = 100000
graph = knight_graph BOARD_WIDTH
tours = graph_tours graph, MAX_NUM_SOLUTIONS
console.log "#{tours.length} tours found (showing first and last)"
illustrate_knights_tour tours[0], BOARD_WIDTH
illustrate_knights_tour tours.pop(), BOARD_WIDTH
output
> time coffee knight.coffee
100000 tours found (showing first and last)
------
1 4 57 20 47 6 49 22
34 19 2 5 58 21 46 7
3 56 35 60 37 48 23 50
18 33 38 55 52 59 8 45
39 14 53 36 61 44 51 24
32 17 40 43 54 27 62 9
13 42 15 30 11 64 25 28
16 31 12 41 26 29 10 63
------
1 4 41 20 63 6 61 22
34 19 2 5 42 21 44 7
3 40 35 64 37 62 23 60
18 33 38 47 56 43 8 45
39 14 57 36 49 46 59 24
32 17 48 55 58 27 50 9
13 54 15 30 11 52 25 28
16 31 12 53 26 29 10 51
real 0m29.741s
user 0m25.656s
sys 0m0.253s
D
Fast Version
import std.stdio, std.algorithm, std.random, std.range,
std.conv, std.typecons, std.typetuple;
int[N][N] knightTour(size_t N=8)(in string start)
in {
assert(start.length >= 2);
} body {
static struct P { int x, y; }
immutable P[8] moves = [P(2,1), P(1,2), P(-1,2), P(-2,1),
P(-2,-1), P(-1,-2), P(1,-2), P(2,-1)];
int[N][N] data;
int[8] sortMoves(in int x, in int y) {
int[2][8] counts;
foreach (immutable i, immutable ref d1; moves) {
int c = 0;
foreach (immutable ref d2; moves) {
immutable p = P(x + d1.x + d2.x, y + d1.y + d2.y);
if (p.x >= 0 && p.x < N && p.y >= 0 && p.y < N &&
data[p.y][p.x] == 0)
c++;
}
counts[i] = [c, i];
}
counts[].randomShuffle; // Shuffle to randomly break ties.
counts[].sort(); // Lexicographic sort.
int[8] result = void;
transversal(counts[], 1).copy(result[]);
return result;
}
immutable p0 = P(start[0] - 'a', N - to!int(start[1 .. $]));
data[p0.y][p0.x] = 1;
Tuple!(int, int, int, int[8])[N * N] order;
order[0] = tuple(p0.x, p0.y, 0, sortMoves(p0.x, p0.y));
int n = 0;
while (n < (N * N - 1)) {
immutable int x = order[n][0];
immutable int y = order[n][1];
bool ok = false;
foreach (immutable i; order[n][2] .. 8) {
immutable P d = moves[order[n][3][i]];
if (x+d.x < 0 || x+d.x >= N || y+d.y < 0 || y+d.y >= N)
continue;
if (data[y + d.y][x + d.x] == 0) {
order[n][2] = i + 1;
n++;
data[y + d.y][x + d.x] = n + 1;
order[n] = tuple(x+d.x,y+d.y,0,sortMoves(x+d.x,y+d.y));
ok = true;
break;
}
}
if (!ok) { // Failed. Backtrack.
data[y][x] = 0;
n--;
}
}
return data;
}
void main() {
foreach (immutable i, side; TypeTuple!(5, 8, 31, 101)) {
immutable form = "%(%" ~ text(side ^^ 2).length.text ~ "d %)";
foreach (ref row; ["c3", "b5", "a1", "a1"][i].knightTour!side)
writefln(form, row);
writeln();
}
}
- Output:
23 16 11 6 21 10 5 22 17 12 15 24 1 20 7 4 9 18 13 2 25 14 3 8 19 63 20 3 24 59 36 5 26 2 23 64 37 4 25 58 35 19 62 21 50 55 60 27 6 22 1 54 61 38 45 34 57 53 18 49 44 51 56 7 28 12 15 52 39 46 31 42 33 17 48 13 10 43 40 29 8 14 11 16 47 30 9 32 41 275 112 19 116 277 604 21 118 823 770 23 120 961 940 25 122 943 926 27 124 917 898 29 126 911 872 31 128 197 870 33 18 115 276 601 20 117 772 767 22 119 958 851 24 121 954 941 26 123 936 925 28 125 912 899 30 127 910 871 32 129 198 111 274 113 278 605 760 603 822 771 824 769 948 957 960 939 944 953 942 927 916 929 918 897 908 913 900 873 196 875 34 869 114 17 600 273 602 775 766 773 768 949 850 959 852 947 952 955 932 937 930 935 924 915 920 905 894 909 882 901 868 199 130 271 110 279 606 759 610 761 776 821 764 825 816 951 956 853 938 945 934 923 928 919 896 893 914 907 904 867 874 195 876 35 16 581 272 599 280 607 774 765 762 779 950 849 826 815 946 933 854 931 844 857 890 921 906 895 886 883 902 881 200 131 194 109 270 281 580 609 758 611 744 777 820 763 780 817 848 827 808 811 846 855 922 843 858 889 892 903 866 885 192 877 36 201 282 15 582 269 598 579 608 757 688 745 778 819 754 783 814 847 828 807 810 845 856 891 842 859 884 887 880 863 202 193 132 267 108 283 578 583 612 689 614 743 756 691 746 781 818 753 784 809 812 829 806 801 840 835 888 865 862 203 878 191 530 37 14 569 268 585 284 597 576 619 690 687 742 755 692 747 782 813 752 785 802 793 830 805 860 841 836 879 864 529 204 133 190 107 266 285 570 577 584 613 686 615 620 695 684 741 732 711 748 739 794 751 786 803 800 839 834 861 528 837 188 531 38 205 286 13 568 265 586 575 596 591 618 685 616 655 696 693 740 733 712 749 738 795 792 831 804 799 838 833 722 527 206 189 134 263 106 287 508 571 590 587 574 621 592 639 694 683 656 731 710 715 734 787 750 737 796 791 832 721 798 207 532 187 474 39 12 417 264 567 288 509 572 595 588 617 654 657 640 697 680 713 730 709 716 735 788 727 720 797 790 723 526 473 208 135 186 105 262 289 416 507 566 589 512 573 622 593 638 653 682 659 698 679 714 729 708 717 736 789 726 719 472 533 184 475 40 209 290 11 418 261 502 415 510 565 594 513 562 641 658 637 652 681 660 699 678 669 728 707 718 675 724 525 704 471 210 185 136 259 104 291 414 419 506 503 514 511 564 623 548 561 642 551 636 651 670 661 700 677 674 725 706 703 534 211 476 183 396 41 10 331 260 493 292 501 420 495 504 515 498 563 624 549 560 643 662 635 650 671 668 701 676 673 524 705 470 395 212 137 182 103 258 293 330 413 494 505 500 455 496 547 516 485 552 625 550 559 644 663 634 649 672 667 702 535 394 477 180 397 42 213 294 9 332 257 492 329 456 421 490 499 458 497 546 517 484 553 626 543 558 645 664 633 648 523 666 469 536 393 220 181 138 255 102 295 328 333 412 491 438 457 454 489 440 459 486 545 518 483 554 627 542 557 646 665 632 537 478 221 398 179 214 43 8 319 256 335 296 345 326 409 422 439 436 453 488 441 460 451 544 519 482 555 628 541 522 647 468 631 392 219 222 139 178 101 254 297 320 327 334 411 346 437 408 423 368 435 452 487 442 461 450 445 520 481 556 629 538 479 466 399 176 215 44 165 298 7 318 253 336 325 344 349 410 347 360 407 424 383 434 427 446 443 462 449 540 521 480 467 630 391 218 223 164 177 140 251 100 303 300 321 316 337 324 343 350 369 382 367 406 425 384 433 428 447 444 463 430 539 390 465 400 175 216 169 166 45 6 299 252 317 304 301 322 315 348 361 342 359 370 381 366 405 426 385 432 429 448 389 464 401 174 217 224 163 150 141 168 99 250 241 302 235 248 307 338 323 314 351 362 341 358 371 380 365 404 377 386 431 402 173 388 225 160 153 170 167 46 143 240 5 98 249 242 305 234 247 308 339 232 313 352 363 230 357 372 379 228 403 376 387 226 159 154 171 162 149 142 151 82 63 2 239 66 97 236 243 306 233 246 309 340 231 312 353 364 229 356 373 378 227 158 375 172 161 148 155 152 83 144 47 4 67 64 61 238 69 96 59 244 71 94 57 310 73 92 55 354 75 90 53 374 77 88 51 156 79 86 49 146 81 84 1 62 3 68 65 60 237 70 95 58 245 72 93 56 311 74 91 54 355 76 89 52 157 78 87 50 147 80 85 48 145
Shorter Version
import std.stdio, std.math, std.algorithm, std.range, std.typecons;
alias Square = Tuple!(int,"x", int,"y");
const(Square)[] knightTour(in Square[] board, in Square[] moves) pure @safe nothrow {
enum findMoves = (in Square sq) pure nothrow @safe =>
cartesianProduct([1, -1, 2, -2], [1, -1, 2, -2])
.filter!(ij => ij[0].abs != ij[1].abs)
.map!(ij => Square(sq.x + ij[0], sq.y + ij[1]))
.filter!(s => board.canFind(s) && !moves.canFind(s));
auto newMoves = findMoves(moves.back);
if (newMoves.empty)
return moves;
//alias warnsdorff = min!(s => findMoves(s).walkLength);
//immutable newSq = newMoves.dropOne.fold!warnsdorff(newMoves.front);
auto pairs = newMoves.map!(s => tuple(findMoves(s).walkLength, s));
immutable newSq = reduce!min(pairs.front, pairs.dropOne)[1];
return board.knightTour(moves ~ newSq);
}
void main(in string[] args) {
enum toSq = (in string xy) => Square(xy[0] - '`', xy[1] - '0');
immutable toAlg = (in Square s) => [dchar(s.x + '`'), dchar(s.y + '0')];
immutable sq = toSq((args.length == 2) ? args[1] : "e5");
const board = iota(1, 9).cartesianProduct(iota(1, 9)).map!Square.array;
writefln("%(%-(%s -> %)\n%)", board.knightTour([sq]).map!toAlg.chunks(8));
}
- Output:
e5 -> d7 -> b8 -> a6 -> b4 -> a2 -> c1 -> b3 a1 -> c2 -> a3 -> b1 -> d2 -> f1 -> h2 -> g4 h6 -> g8 -> e7 -> c8 -> a7 -> c6 -> a5 -> b7 d8 -> f7 -> h8 -> g6 -> f8 -> h7 -> f6 -> e8 g7 -> h5 -> g3 -> h1 -> f2 -> d1 -> b2 -> a4 b6 -> a8 -> c7 -> b5 -> c3 -> d5 -> e3 -> c4 d6 -> e4 -> c5 -> d3 -> e1 -> g2 -> h4 -> f5 d4 -> e2 -> f4 -> e6 -> g5 -> f3 -> g1 -> h3
Delphi
Brute force method. Takes a long time for most solutions, so some optimization should be used. However, it has nice graphics.
{ These routines would normally be in a library,
but are presented here for clarity }
function PointAdd(V1,V2: TPoint): TPoint;
{Add V1 and V2}
begin
Result.X:= V1.X+V2.X;
Result.Y:= V1.Y+V2.Y;
end;
const KnightMoves: array [0..7] of TPoint = (
(X: 2; Y:1),(X: 2; Y:-1),
(X:-2; Y:1),(X:-2; Y:-1),
(X:1; Y: 2),(X:-1; Y: 2),
(X:1; Y:-2),(X:-1; Y:-2));
var Board: array [0..7,0..7] of boolean;
var Path: array of TPoint;
var CellSize,BoardSize: integer;
var CurPos: TPoint;
var BestPath: integer;
{-------------------------------------------------------------}
procedure DrawBestPath(Image: TImage);
begin
Image.Canvas.TextOut(BoardSize+5,5, IntToStr(BestPath));
end;
procedure PushPath(P: TPoint);
begin
SetLength(Path,Length(Path)+1);
Path[High(Path)]:=P;
if Length(Path)>BestPath then BestPath:=Length(Path);
end;
function PopPath: TPoint;
begin
if Length(Path)<1 then exit;
Result:=Path[High(Path)];
SetLength(Path,Length(Path)-1);
end;
procedure ClearPath;
begin
SetLength(Path,0);
end;
{-------- Routines to draw chess board and path --------------}
function GetCellCenter(P: TPoint): TPoint;
{Get pixel position of the center of cell}
begin
Result.X:=CellSize div 2 + CellSize * P.X;
Result.Y:=CellSize div 2 + CellSize * P.Y;
end;
procedure DrawPoint(Canvas: TCanvas; P: TPoint);
{Draw a point on the board}
begin
Canvas.Pen.Color:=clYellow;
Canvas.MoveTo(P.X-1,P.Y-1);
Canvas.LineTo(P.X+1,P.Y+1);
Canvas.MoveTo(P.X+1,P.Y-1);
Canvas.LineTo(P.X-1,P.Y+1);
end;
procedure DrawPathLine(Canvas: TCanvas; P1,P2: TPoint);
{Draw the path line}
var PS1,PS2: TPoint;
begin
PS1:=GetCellCenter(P1);
PS2:=GetCellCenter(P2);
Canvas.Pen.Width:=5;
Canvas.Pen.Color:=clRed;
Canvas.MoveTo(PS1.X,PS1.Y);
Canvas.LineTo(PS2.X,PS2.Y);
DrawPoint(Canvas,PS1);
DrawPoint(Canvas,PS2);
end;
procedure DrawPath(Canvas: TCanvas);
{Draw all points on the path}
var I: integer;
begin
for I:=0 to High(Path)-1 do
begin
DrawPathLine(Canvas, Path[I],Path[I+1]);
end;
end;
procedure DrawBoard(Canvas: TCanvas);
{Draw the chess board}
var R,R2: TRect;
var X,Y: integer;
var Color: TColor;
begin
Canvas.Pen.Color:=clBlack;
R:=Rect(0,0,BoardSize,BoardSize);
Canvas.Rectangle(R);
R:=Rect(0,0,CellSize,CellSize);
for Y:=0 to High(Board[0]) do
for X:=0 to High(Board) do
begin
R2:=R;
if ((X+Y) mod 2)=0 then Color:=clWhite
else Color:=clBlack;
Canvas.Brush.Color:=Color;
OffsetRect(R2,X * CellSize, Y * CellSize);
Canvas.Rectangle(R2);
end;
DrawPath(Canvas);
end;
function AllVisited: boolean;
{Test if all squares have been visit by path}
var X,Y: integer;
begin
Result:=False;
for Y:=0 to High(Board[0]) do
for X:=0 to High(Board) do
if not Board[X,Y] then exit;
Result:=True;
end;
procedure ClearBoard;
{Clear all board positions}
var X,Y: integer;
begin
for Y:=0 to High(Board[0]) do
for X:=0 to High(Board) do
Board[X,Y]:=False;
end;
function IsValidMove(Pos,Move: TPoint): boolean;
{Test if potential move is valid}
var NP: TPoint;
begin
Result:=False;
NP:=PointAdd(Pos,Move);
if (NP.X<0) or (NP.X>High(Board)) or
(NP.Y<0) or (NP.Y>High(Board[0])) then exit;
if Board[NP.X,NP.Y] then exit;
Result:=True;
end;
procedure ConfigureScreen(Image: TImage);
{Configure screen size}
begin
if Image.Width<Image.Height then BoardSize:=Image.Width
else BoardSize:=Image.Height;
CellSize:=BoardSize div 8;
end;
procedure SetPosition(Image: TImage; P: TPoint; Value: boolean);
{Set a new position by adding it to path}
{Marking position as used and redrawing board}
begin
if Value then PushPath(P)
else P:=PopPath;
Board[P.X,P.Y]:=Value;
DrawBoard(Image.Canvas);
DrawBestPath(Image);
Image.Repaint;
end;
procedure TryAllMoves(Image: TImage; Pos: TPoint);
{Recursively try all moves}
var I: integer;
var NewPos: TPoint;
begin
SetPosition(Image,Pos,True);
if AllVisited then exit;
for I:=0 to High(KnightMoves) do
begin
if AbortFlag then Exit;
if IsValidMove(Pos,KnightMoves[I]) then
begin
NewPos:=PointAdd(Pos,KnightMoves[I]);
TryAllMoves(Image,NewPos);
end;
end;
SetPosition(Image,Pos,False);
Application.ProcessMessages;
end;
procedure DoKnightsTour(Image: TImage);
{Solve Knights tour by testing all paths}
begin
BestPath:=0;
ConfigureScreen(Image);
ClearPath;
ClearBoard;
DrawBoard(Image.Canvas);
TryAllMoves(Image, Point(0,0));
end;
- Output:
EasyLang
Backtracking, whereby the paths with the fewest branches are tracked first.
sys topleft
dirs[][] = [ [ 1 2 ] [ 1 -2 ] [ 2 1 ] [ 2 -1 ] [ -1 2 ] [ -1 -2 ] [ -2 -1 ] [ -2 1 ] ]
global brd[][] size .
func cntmoves m[] .
for d = 1 to len dirs[][]
rn = m[1] + dirs[d][1]
cn = m[2] + dirs[d][2]
if rn >= 1 and rn <= size and cn >= 1 and cn <= size and brd[rn][cn] = 0
n += 1
.
.
return n
.
proc sortmoves . m[][] .
for i = 1 to len m[][]
cnt[] &= cntmoves m[i][]
.
for i = 1 to len cnt[] - 1
for j = i + 1 to len cnt[]
if cnt[j] < cnt[i]
swap cnt[j] cnt[i]
swap m[j][] m[i][]
.
.
.
.
func solve r c cnt .
if cnt > size * size
return 1
.
movs[][] = [ ]
for d = 1 to len dirs[][]
rn = r + dirs[d][1]
cn = c + dirs[d][2]
if rn >= 1 and rn <= size and cn >= 1 and cn <= size and brd[rn][cn] = 0
movs[][] &= [ rn cn ]
.
.
sortmoves movs[][]
for i = 1 to len movs[][]
rn = movs[i][1]
cn = movs[i][2]
brd[rn][cn] = cnt
if solve rn cn (cnt + 1) = 1
return 1
.
brd[rn][cn] = 0
.
return 0
.
proc prepare . .
brd[][] = [ ]
len brd[][] size
for r to size
len brd[r][] size
.
.
proc printbrd . .
numfmt 0 3
if size > 10
numfmt 0 4
.
for r to size
for c to size
write brd[r][c]
.
print ""
.
.
size = 8
r0 = random size
c0 = random size
prepare
brd[r0][c0] = 1
found = solve r0 c0 2
if found = 1
printbrd
print ""
else
print "no solutions found: (" & r0 & " " & c0 & ")"
.
#
proc showgraf . .
sc = 100 / size
linewidth sc / 15
col[] = [ 777 333 ]
for r = 0 to size - 1
for c = 0 to size - 1
color col[(r + c) mod1 2]
move sc * c sc * r
rect sc sc
.
.
move 0 / 0 0
for i to size * size
for r = 1 to size
for c = 1 to size
if brd[r][c] = i
color 600
x = c * sc - sc / 2
y = r * sc - sc / 2
line x y
color 880
circle sc / 10
.
.
.
.
.
if found = 1
showgraf
.
- Output:
28 25 10 57 38 23 8 5 11 58 27 24 9 6 37 22 26 29 60 45 56 39 4 7 59 12 55 52 41 44 21 36 30 53 42 61 46 51 40 3 13 62 47 54 43 18 35 20 48 31 64 15 50 33 2 17 63 14 49 32 1 16 19 34
EchoLisp
The algorithm uses iterative backtracking and Warnsdorff's heuristic. It can output closed or non-closed tours.
(require 'plot)
(define *knight-moves*
'((2 . 1)(2 . -1 ) (1 . -2) (-1 . -2 )(-2 . -1) (-2 . 1) (-1 . 2) (1 . 2)))
(define *hit-squares* null)
(define *legal-moves* null)
(define *tries* 0)
(define (square x y n ) (+ y (* x n)))
(define (dim n) (1- (* n n))) ; n^2 - 1
;; check legal knight move from sq
;; return null or (list destination-square)
(define (legal-disp n sq k-move)
(let ((x (+ (quotient sq n) (first k-move)))
(y (+ (modulo sq n) (rest k-move))))
(if (and (>= x 0) (< x n) (>= y 0) (< y n))
(list (square x y n)) null)))
;; list of legal destination squares from sq
(define (legal-moves sq k-moves n )
(if (null? k-moves) null
(append (legal-moves sq (rest k-moves) n) (legal-disp n sq (first k-moves)))))
;; square freedom = number of destination squares not already reached
(define (freedom sq)
(for/sum ((dest (vector-ref *legal-moves* sq)))
(if (vector-ref *hit-squares* dest) 0 1)))
;; The chess adage" A knight on the rim is dim" is false here :
;; choose to move to square with smallest freedom : Warnsdorf's rule
(define (square-sort a b)
(< (freedom a) (freedom b)))
;; knight tour engine
(define (play sq step starter last-one wants-open)
(set! *tries* (1+ *tries*))
(vector-set! *hit-squares* sq step) ;; flag used square
(if (= step last-one) (throw 'HIT last-one)) ;; stop on first path found
(when (or wants-open ;; cut search iff closed path
(and (< step last-one) (> (freedom starter) 0))) ;; this ensures a closed path
(for ((target (list-sort square-sort (vector-ref *legal-moves* sq))))
(unless (vector-ref *hit-squares* target)
(play target (1+ step) starter last-one wants-open))))
(vector-set! *hit-squares* sq #f)) ;; unflag used square
(define (show-steps n wants-open)
(string-delimiter "")
(if wants-open
(printf "♘-tour: %d tries." *tries*)
(printf "♞-closed-tour: %d tries." *tries*))
(for ((x n))
(writeln)
(for((y n))
(write (string-pad-right (vector-ref *hit-squares* (square x y n)) 4)))))
(define (k-tour (n 8) (starter 0) (wants-open #t))
(set! *hit-squares* (make-vector (* n n) #f))
;; build vector of legal moves for squares 0..n^2-1
(set! *legal-moves*
(build-vector (* n n) (lambda(sq) (legal-moves sq *knight-moves* n))))
(set! *tries* 0) ; counter
(try
(play starter 0 starter (dim n) wants-open)
(catch (hit mess) (show-steps n wants-open))))
- Output:
(k-tour 8 0 #f)
♞-closed-tour: 66 tries.
0 47 14 31 62 27 12 29
15 32 63 54 13 30 57 26
48 1 46 61 56 59 28 11
33 16 55 50 53 44 25 58
2 49 42 45 60 51 10 39
17 34 19 52 43 40 7 24
20 3 36 41 22 5 38 9
35 18 21 4 37 8 23 6
(k-tour 20 57)
♘-tour: 400 tries.
31 34 29 104 209 36 215 300 211 38 213 354 343 40 345 386 383 42 1 388
28 103 32 35 216 299 210 37 214 335 342 39 346 385 382 41 390 387 396 43
33 30 105 208 201 308 301 336 323 212 353 340 355 344 391 384 395 0 389 2
102 27 202 219 298 217 322 309 334 341 356 347 358 351 376 381 378 399 44 397
203 106 207 200 307 228 311 302 337 324 339 352 373 364 379 392 375 394 3 368
26 101 220 229 218 297 304 321 310 333 348 357 350 359 374 377 380 367 398 45
107 204 199 206 227 306 231 312 303 338 325 330 363 372 365 328 393 254 369 4
100 25 122 221 230 233 296 305 320 313 332 349 326 329 360 371 366 251 46 253
121 108 205 198 145 226 237 232 295 286 319 314 331 362 327 316 255 370 5 178
24 99 144 123 222 129 234 279 236 281 294 289 318 315 256 361 250 179 252 47
109 120 111 130 197 146 225 238 285 278 287 272 293 290 317 180 257 162 177 6
98 23 124 143 128 223 276 235 280 239 282 291 288 265 270 249 176 181 48 161
115 110 119 112 131 196 147 224 277 284 273 266 271 292 245 258 163 174 7 58
22 97 114 125 142 127 140 275 194 267 240 283 264 269 248 175 182 59 160 49
87 116 95 118 113 132 195 148 187 274 263 268 191 244 259 246 173 164 57 8
96 21 88 133 126 141 150 139 262 193 190 241 260 247 172 183 60 159 50 65
77 86 117 94 89 138 135 188 149 186 261 192 171 184 243 156 165 64 9 56
20 81 78 85 134 93 90 151 136 189 170 185 242 155 166 61 158 53 66 51
79 76 83 18 91 74 137 16 169 72 153 14 167 70 157 12 63 68 55 10
82 19 80 75 84 17 92 73 152 15 168 71 154 13 62 69 54 11 52 67
- Plotting
64 shades of gray. We plot the move sequence in shades of gray, from black to white. The starting square is red. The ending square is green. One can observe that the squares near the border are played first (dark squares).
(define (step-color x y n last-one)
(letrec ((sq (square (floor x) (floor y) n))
(step (vector-ref *hit-squares* sq) n n))
(cond ((= 0 step) (rgb 1 0 0)) ;; red starter
((= last-one step) (rgb 0 1 0)) ;; green end
(else (gray (// step n n))))))
(define ( k-plot n)
(plot-rgb (lambda (x y) (step-color x y n (dim n))) (- n epsilon) (- n epsilon)))
Closed path on a 12x12 board: [1]
![[1]](http://www.echolalie.org/echolisp/images/k-tour-12.png){kind=link}
Open path on a 24x24 board: [2]
![[2]](http://www.echolalie.org/echolisp/images/k-tour-24.png){kind=link}
Elixir
defmodule Board do
import Integer, only: [is_odd: 1]
defmodule Cell do
defstruct [:value, :adj]
end
@adjacent [[-1,-2],[-2,-1],[-2,1],[-1,2],[1,2],[2,1],[2,-1],[1,-2]]
defp initialize(rows, cols) do
board = for i <- 1..rows, j <- 1..cols, into: %{}, do: {{i,j}, true}
for i <- 1..rows, j <- 1..cols, into: %{} do
adj = for [di,dj] <- @adjacent, board[{i+di, j+dj}], do: {i+di, j+dj}
{{i,j}, %Cell{value: 0, adj: adj}}
end
end
defp solve(board, ij, num, goal) do
board = Map.update!(board, ij, fn cell -> %{cell | value: num} end)
if num == goal do
throw({:ok, board})
else
wdof(board, ij)
|> Enum.each(fn k -> solve(board, k, num+1, goal) end)
end
end
defp wdof(board, ij) do # Warnsdorf's rule
board[ij].adj
|> Enum.filter(fn k -> board[k].value == 0 end)
|> Enum.sort_by(fn k ->
Enum.count(board[k].adj, fn x -> board[x].value == 0 end)
end)
end
defp to_string(board, rows, cols) do
width = to_string(rows * cols) |> String.length
format = String.duplicate("~#{width}w ", cols)
Enum.map_join(1..rows, "\n", fn i ->
:io_lib.fwrite format, (for j <- 1..cols, do: board[{i,j}].value)
end)
end
def knight_tour(rows, cols, sx, sy) do
IO.puts "\nBoard (#{rows} x #{cols}), Start: [#{sx}, #{sy}]"
if is_odd(rows*cols) and is_odd(sx+sy) do
IO.puts "No solution"
else
try do
initialize(rows, cols)
|> solve({sx,sy}, 1, rows*cols)
IO.puts "No solution"
catch
{:ok, board} -> IO.puts to_string(board, rows, cols)
end
end
end
end
Board.knight_tour(8,8,4,2)
Board.knight_tour(5,5,3,3)
Board.knight_tour(4,9,1,1)
Board.knight_tour(5,5,1,2)
Board.knight_tour(12,12,2,2)
- Output:
Board (8 x 8), Start: [4, 2] 23 20 3 32 25 10 5 8 2 35 24 21 4 7 26 11 19 22 33 36 31 28 9 6 34 1 50 29 48 37 12 27 51 18 53 44 61 30 47 38 54 43 56 49 58 45 62 13 17 52 41 60 15 64 39 46 42 55 16 57 40 59 14 63 Board (5 x 5), Start: [3, 3] 19 8 3 14 25 2 13 18 9 4 7 20 1 24 15 12 17 22 5 10 21 6 11 16 23 Board (4 x 9), Start: [1, 1] 1 34 3 28 13 24 9 20 17 4 29 6 33 8 27 18 23 10 35 2 31 14 25 12 21 16 19 30 5 36 7 32 15 26 11 22 Board (5 x 5), Start: [1, 2] No solution Board (12 x 12), Start: [2, 2] 87 24 59 2 89 26 61 4 39 8 31 6 58 1 88 25 60 3 92 27 30 5 38 9 23 86 83 90 103 98 29 62 93 40 7 32 82 57 102 99 84 91 104 97 28 37 10 41 101 22 85 114 105 116 111 94 63 96 33 36 56 81 100 123 128 113 106 117 110 35 42 11 21 122 141 80 115 124 127 112 95 64 109 34 144 55 78 121 142 129 118 107 126 133 12 43 51 20 143 140 79 120 125 138 69 108 65 134 54 73 52 77 130 139 70 119 132 137 44 13 19 50 75 72 17 48 131 68 15 46 135 66 74 53 18 49 76 71 16 47 136 67 14 45
Elm
module Main exposing (main)
import Browser exposing (element)
import Html as H
import Html.Attributes as HA
import List exposing (filter, head, length, map, map2, member, tail)
import List.Extra exposing (andThen, minimumBy)
import String exposing (join)
import Svg exposing (g, line, rect, svg)
import Svg.Attributes exposing (fill, height, style, version, viewBox, width, x, x1, x2, y, y1, y2)
import Svg.Events exposing (onClick)
import Time exposing (every)
import Tuple
type alias Cell =
( Int, Int )
type alias BoardSize =
( Int, Int )
type alias Model =
{ path : List Cell
, board : List Cell
, pause_ms : Float
, size : BoardSize
}
type Msg
= Tick Time.Posix
| SetStart Cell
| SetSize BoardSize
| SetPause Float
boardsize_width: BoardSize -> Int
boardsize_width bs =
Tuple.second bs
boardsize_height: BoardSize -> Int
boardsize_height bs =
Tuple.first bs
boardsize_dec: Int -> Int
boardsize_dec n =
let
minimum_size = 3
in
if n <= minimum_size then
minimum_size
else
n - 1
boardsize_inc: Int -> Int
boardsize_inc n =
let
maximum_size = 40
in
if n >= maximum_size then
maximum_size
else
n + 1
pause_inc: Float -> Float
pause_inc n =
n + 10
-- decreasing pause time (ms) increases speed
pause_dec: Float -> Float
pause_dec n =
let
minimum_pause = 0
in
if n <= minimum_pause then
minimum_pause
else
n - 10
board_init : BoardSize -> List Cell
board_init board_size =
List.range 0 (boardsize_height board_size - 1)
|> andThen
(\r ->
List.range 0 (boardsize_width board_size - 1)
|> andThen
(\c ->
[ ( r, c ) ]
)
)
nextMoves : Model -> Cell -> List Cell
nextMoves model ( stRow, stCol ) =
let
c =
[ 1, 2, -1, -2 ]
km =
c
|> andThen
(\cRow ->
c
|> andThen
(\cCol ->
if abs cRow == abs cCol then
[]
else
[ ( cRow, cCol ) ]
)
)
jumps =
List.map (\( kmRow, kmCol ) -> ( kmRow + stRow, kmCol + stCol )) km
in
List.filter (\j -> List.member j model.board && not (List.member j model.path)) jumps
bestMove : Model -> Maybe Cell
bestMove model =
case List.head model.path of
Just mph ->
minimumBy (List.length << nextMoves model) (nextMoves model mph)
_ ->
Nothing
-- Initialize the application - https://guide.elm-lang.org/effects/
init : () -> ( Model, Cmd Msg )
init _ =
let
-- Initial board height and width
initial_size =
8
-- Initial chess board
initial_board =
board_init (initial_size, initial_size)
initial_path =
[]
initial_pause =
10
in
( Model initial_path initial_board initial_pause (initial_size, initial_size), Cmd.none )
-- View the model - https://guide.elm-lang.org/effects/
view : Model -> H.Html Msg
view model =
let
showChecker row col =
rect
[ x <| String.fromInt col
, y <| String.fromInt row
, width "1"
, height "1"
, fill <|
if modBy 2 (row + col) == 0 then
"blue"
else
"grey"
, onClick <| SetStart ( row, col )
]
[]
showMove ( row0, col0 ) ( row1, col1 ) =
line
[ x1 <| String.fromFloat (toFloat col0 + 0.5)
, y1 <| String.fromFloat (toFloat row0 + 0.5)
, x2 <| String.fromFloat (toFloat col1 + 0.5)
, y2 <| String.fromFloat (toFloat row1 + 0.5)
, style "stroke:yellow;stroke-width:0.05"
]
[]
render mdl =
let
checkers =
mdl.board
|> andThen
(\( r, c ) ->
[ showChecker r c ]
)
moves =
case List.tail mdl.path of
Nothing ->
[]
Just tl ->
List.map2 showMove mdl.path tl
in
checkers ++ moves
unvisited =
length model.board - length model.path
center =
[ HA.style "text-align" "center" ]
table =
[ HA.style "text-align" "center", HA.style "display" "table", HA.style "width" "auto", HA.style "margin" "auto" ]
table_row =
[ HA.style "display" "table-row", HA.style "width" "auto" ]
table_cell =
[ HA.style "display" "table-cell", HA.style "width" "auto", HA.style "padding" "1px 3px" ]
rows =
boardsize_height model.size
cols =
boardsize_width model.size
in
H.div
[]
[ H.h1 center [ H.text "Knight's Tour" ]
-- controls
, H.div
table
[ H.div -- labels
table_row
[ H.div
table_cell
[ H.text "Rows"]
, H.div
table_cell
[ H.text "Columns"]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.text "Pause (ms)"]
]
, H.div
table_row
[ H.div -- Increase
table_cell
[ H.button [onClick <| SetSize ( boardsize_inc rows, cols )] [ H.text "▲"] ]
, H.div
table_cell
[ H.button [onClick <| SetSize ( rows, boardsize_inc cols )] [ H.text "▲"] ]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.button [onClick <| SetPause ( pause_inc model.pause_ms )] [ H.text "▲"] ]
]
, H.div
table_row
[ H.div -- Value
table_cell
[ H.text <| String.fromInt rows ]
, H.div
table_cell
[ H.text <| String.fromInt cols]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.text <| String.fromFloat model.pause_ms]
]
, H.div
table_row
[ H.div -- Decrease
table_cell
[ H.button [onClick <| SetSize ( boardsize_dec rows, cols )] [ H.text "▼"] ]
, H.div
table_cell
[ H.button [onClick <| SetSize ( rows, boardsize_dec cols )] [ H.text "▼"] ]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.button [onClick <| SetPause ( pause_dec model.pause_ms )] [ H.text "▼"] ]
]
]
, H.h2 center [ H.text "(pick a square)" ]
, H.div -- chess board
center
[ svg
[ version "1.1"
, width (String.fromInt (25 * cols))
, height (String.fromInt (25 * rows))
, viewBox
(join " "
[ String.fromInt 0
, String.fromInt 0
, String.fromInt cols
, String.fromInt rows
]
)
]
[ g [] <| render model ]
]
, H.h3 center [ H.text <| "Unvisited count : " ++ String.fromInt unvisited ]
]
-- Update the model - https://guide.elm-lang.org/effects/
update : Msg -> Model -> ( Model, Cmd Msg )
update msg model =
let
mo =
case msg of
SetPause pause ->
{ model | pause_ms = pause }
SetSize board_size ->
{ model | board = board_init board_size, path = [], size = board_size }
SetStart start ->
{ model | path = [ start ] }
Tick _ ->
case model.path of
[] ->
model
_ ->
case bestMove model of
Nothing ->
model
Just best ->
{ model | path = best :: model.path }
in
( mo, Cmd.none )
-- Subscribe to https://guide.elm-lang.org/effects/
subscriptions : Model -> Sub Msg
subscriptions model =
Time.every model.pause_ms Tick
-- Application entry point
main: Program () Model Msg
main =
element -- https://package.elm-lang.org/packages/elm/browser/latest/Browser#element
{ init = init
, view = view
, update = update
, subscriptions = subscriptions
}
Link to live demo: https://dmcbane.github.io/knights-tour/
Erlang
Again I use backtracking. It seemed easier this time.
-module( knights_tour ).
-export( [display/1, solve/1, task/0] ).
display( Moves ) ->
%% The knigh walks the moves {Position, Step_nr} order.
%% Top left corner is {$a, 8}, Bottom right is {$h, 1}.
io:fwrite( "Moves:" ),
lists:foldl( fun display_moves/2, erlang:length(Moves), lists:keysort(2, Moves) ),
io:nl(),
[display_row(Y, Moves) || Y <- lists:seq(8, 1, -1)].
solve( First_square ) ->
try
bt_loop( 1, next_moves(First_square), [{First_square, 1}] )
catch
_:{ok, Moves} -> Moves
end.
task() ->
io:fwrite( "Starting {a, 1}~n" ),
Moves = solve( {$a, 1} ),
display( Moves ).
bt( N, Move, Moves ) -> bt_reject( is_not_allowed_knight_move(Move, Moves), N, Move, [{Move, N} | Moves] ).
bt_accept( true, _N, _Move, Moves ) -> erlang:throw( {ok, Moves} );
bt_accept( false, N, Move, Moves ) -> bt_loop( N, next_moves(Move), Moves ).
bt_loop( N, New_moves, Moves ) -> [bt( N+1, X, Moves ) || X <- New_moves].
bt_reject( true, _N, _Move, _Moves ) -> backtrack;
bt_reject( false, N, Move, Moves ) -> bt_accept( is_all_knights(Moves), N, Move, Moves ).
display_moves( {{X, Y}, 1}, Max ) ->
io:fwrite(" ~p. N~c~p", [1, X, Y]),
Max;
display_moves( {{X, Y}, Max}, Max ) ->
io:fwrite(" N~c~p~n", [X, Y]),
Max;
display_moves( {{X, Y}, Step_nr}, Max ) when Step_nr rem 8 =:= 0 ->
io:fwrite(" N~c~p~n~p. N~c~p", [X, Y, Step_nr, X, Y]),
Max;
display_moves( {{X, Y}, Step_nr}, Max ) ->
io:fwrite(" N~c~p ~p. N~c~p", [X, Y, Step_nr, X, Y]),
Max.
display_row( Row, Moves ) ->
[io:fwrite(" ~2b", [proplists:get_value({X, Row}, Moves)]) || X <- [$a, $b, $c, $d, $e, $f, $g, $h]],
io:nl().
is_all_knights( Moves ) when erlang:length(Moves) =:= 64 -> true;
is_all_knights( _Moves ) -> false.
is_asymetric( Start_column, Start_row, Stop_column, Stop_row ) ->
erlang:abs( Start_column - Stop_column ) =/= erlang:abs( Start_row - Stop_row ).
is_not_allowed_knight_move( Move, Moves ) ->
no_such_move =/= proplists:get_value( Move, Moves, no_such_move ).
next_moves( {Column, Row} ) ->
[{X, Y} || X <- next_moves_column(Column), Y <- next_moves_row(Row), is_asymetric(Column, Row, X, Y)].
next_moves_column( $a ) -> [$b, $c];
next_moves_column( $b ) -> [$a, $c, $d];
next_moves_column( $g ) -> [$e, $f, $h];
next_moves_column( $h ) -> [$g, $f];
next_moves_column( C ) -> [C - 2, C - 1, C + 1, C + 2].
next_moves_row( 1 ) -> [2, 3];
next_moves_row( 2 ) -> [1, 3, 4];
next_moves_row( 7 ) -> [5, 6, 8];
next_moves_row( 8 ) -> [6, 7];
next_moves_row( N ) -> [N - 2, N - 1, N + 1, N + 2].
- Output:
17> knights_tour:task().
Starting {a, 1}
Moves: 1. Na1 Nb3 2. Nb3 Na5 3. Na5 Nb7 4. Nb7 Nc5 5. Nc5 Na4 6. Na4 Nb2 7. Nb2 Nc4
8. Nc4 Na3 9. Na3 Nb1 10. Nb1 Nc3 11. Nc3 Na2 12. Na2 Nb4 13. Nb4 Na6 14. Na6 Nb8 15. Nb8 Nc6
16. Nc6 Na7 17. Na7 Nb5 18. Nb5 Nc7 19. Nc7 Na8 20. Na8 Nb6 21. Nb6 Nc8 22. Nc8 Nd6 23. Nd6 Ne4
24. Ne4 Nd2 25. Nd2 Nf1 26. Nf1 Ne3 27. Ne3 Nc2 28. Nc2 Nd4 29. Nd4 Ne2 30. Ne2 Nc1 31. Nc1 Nd3
32. Nd3 Ne1 33. Ne1 Ng2 34. Ng2 Nf4 35. Nf4 Nd5 36. Nd5 Ne7 37. Ne7 Ng8 38. Ng8 Nh6 39. Nh6 Nf5
40. Nf5 Nh4 41. Nh4 Ng6 42. Ng6 Nh8 43. Nh8 Nf7 44. Nf7 Nd8 45. Nd8 Ne6 46. Ne6 Nf8 47. Nf8 Nd7
48. Nd7 Ne5 49. Ne5 Ng4 50. Ng4 Nh2 51. Nh2 Nf3 52. Nf3 Ng1 53. Ng1 Nh3 54. Nh3 Ng5 55. Ng5 Nh7
56. Nh7 Nf6 57. Nf6 Ne8 58. Ne8 Ng7 59. Ng7 Nh5 60. Nh5 Ng3 61. Ng3 Nh1 62. Nh1 Nf2 63. Nf2 Nd1
20 15 22 45 58 47 38 43
17 4 19 48 37 44 59 56
14 21 16 23 46 57 42 39
3 18 5 36 49 40 55 60
6 13 8 29 24 35 50 41
9 2 11 32 27 52 61 54
12 7 28 25 30 63 34 51
1 10 31 64 33 26 53 62
ERRE
Taken from ERRE distribution disk. Comments are in Italian.
! **********************************************************************
! * *
! * IL GIRO DEL CAVALLO - come collocare un cavallo su di una *
! * scacchiera n*n passando una sola volta *
! * per ogni casella. *
! * *
! **********************************************************************
! ----------------------------------------------------------------------
! Inizializzazione dei parametri
! ----------------------------------------------------------------------
PROGRAM KNIGHT
!$INTEGER
!$KEY
DIM H[25,25],A[8],B[8],P0[8],P1[8]
!$INCLUDE="PC.LIB"
PROCEDURE INIT_SCACCHIERA
! **********************************************************************
! * Routine di inizializzazione scacchiera *
! **********************************************************************
FOR I1=1 TO 8 DO
U=X+A[I1] V=Y+B[I1]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
H[U,V]=H[U,V]-1
END IF
END FOR
END PROCEDURE
PROCEDURE MOSTRA_SCACCHIERA
! *********************************************************************
! * Routine di visualizzazione della scacchiera *
! *********************************************************************
LOCATE(5,1) COLOR(0,7) PRINT(" Mossa num.";NMOS) COLOR(7,0)
L2=N
FOR I2=1 TO N DO
PRINT
FOR L1=1 TO N DO
IF H[L1,L2]>0 THEN COLOR(15,0) END IF
WRITE("####";H[L1,L2];)
COLOR(7,0)
END FOR
L2=L2-1
END FOR
END PROCEDURE
PROCEDURE AGGIORNA_SCACCHIERA
! *********************************************************************
! * Routine di Aggiornamento Scacchiera *
! *********************************************************************
B=1
FOR I1=1 TO 8 DO
U=X+A[I1] V=Y+B[I1]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 THEN
H[U,V]=H[U,V]+1 B=0
END IF
END IF
END FOR
IF B=1 THEN Q1=0 END IF
END PROCEDURE
PROCEDURE MOSSA_MAX_PESO
! *********************************************************************
! * Cerca la prossima mossa con il massimo peso *
! *********************************************************************
M1=0 RO=1
FOR W=1 TO 8 DO
U=Z1+A[W] V=Z2+B[W]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 AND H[U,V]<=M1 THEN
IF H[U,V]=M1 THEN
RO=RO+1 P0[RO]=W
ELSE
M1=H[U,V] Q1=1 T1=U T2=V RO=1 P0[1]=W
END IF
END IF
END IF
END FOR
END PROCEDURE
PROCEDURE MOSSA_MIN_PESO
! *********************************************************************
! * Cerca la prossima mossa con il minimo peso *
! *********************************************************************
M1=-9 RO=1
FOR W=1 TO 8 DO
U=Z1+A[W] V=Z2+B[W]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 AND H[U,V]>=M1 THEN
IF H[U,V]=M1 THEN
RO=RO+1 P0[RO]=W
ELSE
M1=H[U,V] Q1=1 T1=U T2=V RO=1 P0[1]=W
END IF
END IF
END IF
END FOR
END PROCEDURE
BEGIN
A[1]=1 A[2]=2 A[3]=2 A[4]=1
A[5]=-1 A[6]=-2 A[7]=-2 A[8]=-1
B[1]=2 B[2]=1 B[3]=-1 B[4]=-2
B[5]=-2 B[6]=-1 B[7]=1 B[8]=2
CLS
PRINT(" *** LA GALOPPATA DEL CAVALIERE ***")
PRINT
PRINT("Inserire la dimensione della scacchiera (max. 25)";)
INPUT(N)
PRINT("Inserire la caselle di partenza (x,y) ";)
INPUT(X1,Y1)
NMOS=1 A1=1 N1=N*N ESCAPE=FALSE
! ----------------------------------------------------------------------
! Set della scacchiera
! ----------------------------------------------------------------------
WHILE NOT ESCAPE DO
FOR I=1 TO N DO
FOR J=1 TO N DO
H[I,J]=0
END FOR
END FOR
FOR I=1 TO N DO
FOR J=1 TO N DO
X=I Y=J
INIT_SCACCHIERA
END FOR
END FOR
! ----------------------------------------------------------------------
! Effettua la prima mossa
! ----------------------------------------------------------------------
X=X1 Y=Y1 H[X,Y]=1 L=2
AGGIORNA_SCACCHIERA
Q1=1 Q2=1
! -----------------------------------------------------------------------
! Trova la prossima mossa
! -----------------------------------------------------------------------
WHILE Q1<>0 AND Q2<>0 DO
Q1=0 Z1=X Z2=Y
MOSSA_MIN_PESO
IF RO<=1 THEN
C1=T1 C2=T2
ELSE
! ------------------------------------------------------------------------
! Esamina tutti i vincoli
! ------------------------------------------------------------------------
FOR K=1 TO RO DO
P1[K]=P0[K]
END FOR
R1=RO
IF A1=1 THEN M2=-9 ELSE M2=0 END IF
FOR K=1 TO R1 DO
F1=P1[K] Z1=X+A[F1] Z2=Y+B[F1]
IF A1=1 THEN
MOSSA_MAX_PESO
IF M1<=M2 THEN
!$NULL
ELSE
M2=M1 C1=Z1 C2=Z2
END IF
ELSE
MOSSA_MIN_PESO
IF M1>=M2 THEN
!$NULL
ELSE
M2=M1 C1=Z1 C2=Z2
END IF
END IF
END FOR
! ------------------------------------------------------------------------
! Prossima mossa trovata:aggiorna la scacchiera
! ------------------------------------------------------------------------
END IF
IF Q1<>0 THEN
X=C1 Y=C2 H[X,Y]=L
AGGIORNA_SCACCHIERA
IF L=N1 THEN Q2=0 END IF
END IF
L=L+1
MOSTRA_SCACCHIERA
NMOS=NMOS+1
END WHILE
! ------------------------------------------------------------------------
! La ricerca è terminata: visualizza i risultati
! ------------------------------------------------------------------------
PRINT PRINT
IF Q2<>1 THEN
PRINT("*** Trovata la soluzione! ***")
MOSTRA_SCACCHIERA
ESCAPE=TRUE
ELSE
IF A1=0 THEN
PRINT("Nessuna soluzione.")
ESCAPE=TRUE
ELSE
BEEP
A1=0
END IF
END IF
END WHILE
REPEAT
GET(A$)
UNTIL A$<>""
END PROGRAM- Output:
*** LA GALOPPATA DEL CAVALIERE *** Inserire la dimensione della scacchiera (max. 25)? 8 Inserire la caselle di partenza (x,y) ? 1,1 Mossa num. 64 64 7 54 41 60 9 48 39 53 42 61 8 55 40 35 10 6 63 44 59 34 49 38 47 43 52 21 62 45 56 11 36 20 5 58 33 50 37 46 25 31 2 51 22 57 26 15 12 4 19 32 29 14 17 24 27 1 30 3 18 23 28 13 16 *** Trovata la soluzione! ***
FreeBASIC
Dim Shared As Integer tamano, xc, yc, nm
Dim As Integer f, qm, nmov, n = 0
Dim As String posini
Cls : Color 11
Input "Tamaño tablero: ", tamano
Input "Posicion inicial: ", posini
Dim As Integer x = Asc(Mid(posini,1,1))-96
Dim As Integer y = Val(Mid(posini,2,1))
Dim Shared As Integer tablero(tamano,tamano), dx(8), dy(8)
For f = 1 To 8 : Read dx(f), dy(f) : Next f
Data 2,1,1,2,-1,2,-2,1,-2,-1,-1,-2,1,-2,2,-1
Sub FindMoves()
Dim As Integer i, xt, yt
If xc < 1 Or yc < 1 Or xc > tamano Or yc > tamano Then nm = 1000: Return
If tablero(xc,yc) Then nm = 2000: Return
nm = 0
For i = 1 To 8
xt = xc+dx(i)
yt = yc+dy(i)
If xt < 1 Or yt < 1 Or xt > tamano Or yt > tamano Then 'Salta este movimiento
Elseif tablero(xt,yt) Then 'Salta este movimiento
Else
nm += 1
End If
Next i
End Sub
Color 4, 7 'Pinta tablero
For f = 1 To tamano
Locate 15-tamano, 3*f: Print " "; Chr(96+f); " ";
Locate 17-f, 3*(tamano+1)+1: Print Using "##"; f;
Next f
Color 15, 0
Do
n += 1
tablero(x,y) = n
Locate 17-y, 3*x: Print Using "###"; n;
If n = tamano*tamano Then Exit Do
nmov = 100
For f = 1 To 8
xc = x+dx(f)
yc = y+dy(f)
FindMoves()
If nm < nmov Then nmov = nm: qm = f
Next f
x = x+dx(qm)
y = y+dy(qm)
Sleep 1
Loop
Color 14 : Locate Csrlin+tamano, 1
Print " Pulsa cualquier tecla para finalizar..."
Sleep
End
- Output:
Tamaño tablero: 8 Posicion inicial: c3 a b c d e f g h 24 11 22 19 26 9 38 47 8 21 18 25 10 39 48 27 8 7 12 23 20 53 28 37 46 49 6 17 52 29 40 59 50 7 36 5 30 13 58 51 54 41 62 45 4 57 16 1 42 63 60 35 6 3 2 31 14 55 4 33 44 61 2 15 56 3 32 43 64 5 34 1 Pulsa cualquier tecla para finalizar...
Fōrmulæ
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Programs in Fōrmulæ are created/edited online in its website.
In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.
Fortran
FORTRAN 77
C-----------------------------------------------------------------------
C
C Find Knight’s Tours.
C
C Using Warnsdorff’s heuristic, find multiple solutions.
C Optionally accept only closed tours.
C
C This program is migrated from my implementation for ATS/Postiats.
C Arrays with dimension 1:64 take the place of stack frames.
C
C Compile with, for instance:
C
C gfortran -O2 -g -std=legacy -o knights_tour knights_tour.f
C
C or
C
C f2c knights_tour.f
C cc -O -o knights_tour knights_tour.c -lf2c
C
C Usage examples:
C
C One tour starting at a1, either open or closed:
C
C echo "a1 1 F" | ./knights_tour
C
C No more than 2000 closed tours starting at c5:
C
C echo "c5 2000 T" | ./knights_tour
C
C-----------------------------------------------------------------------
program ktour
implicit none
character*2 alg
integer i, j
integer mxtour
logical closed
read (*,*) alg, mxtour, closed
call alg2ij (alg, i, j)
call explor (i, j, mxtour, closed)
end
C-----------------------------------------------------------------------
subroutine explor (istart, jstart, mxtour, closed)
implicit none
C Explore the space of 'Warnsdorffian' knight’s paths, looking for
C and printing complete tours.
integer istart, jstart ! The starting position.
integer mxtour ! The maximum number of tours to print.
logical closed ! Closed tours only?
integer board(1:8,1:8)
integer imove(1:8,1:64)
integer jmove(1:8,1:64)
integer nmove(1:64)
integer n
integer itours
logical goodmv
logical isclos
itours = 0
call initbd (board)
n = 1
nmove(1) = 8
imove(8, 1) = istart
jmove(8, 1) = jstart
1000 if (itours .lt. mxtour .and. n .ne. 0) then
if (nmove(n) .eq. 9) then
n = n - 1
if (n .ne. 0) then
call unmove (board, imove, jmove, nmove, n)
nmove(n) = nmove(n) + 1
end if
else if (goodmv (imove, nmove, n)) then
call mkmove (board, imove, jmove, nmove, n)
if (n .eq. 64) then
if (.not. closed) then
itours = itours + 1
call prnt (board, itours)
else if (isclos (board)) then
itours = itours + 1
call prnt (board, itours)
end if
call unmove (board, imove, jmove, nmove, n)
nmove(n) = 9
else if (n .eq. 63) then
call possib (board, n, imove, jmove, nmove)
n = n + 1
nmove(n) = 1
else
call nxtmov (board, n, imove, jmove, nmove)
n = n + 1
nmove(n) = 1
end if
else
nmove(n) = nmove(n) + 1
end if
goto 1000
end if
end
C-----------------------------------------------------------------------
subroutine initbd (board)
implicit none
C Initialize a chessboard with empty squares.
integer board(1:8,1:8)
integer i, j
do 1010 j = 1, 8
do 1000 i = 1, 8
board(i, j) = -1
1000 continue
1010 continue
end
C-----------------------------------------------------------------------
subroutine mkmove (board, imove, jmove, nmove, n)
implicit none
C Fill a square with a move number.
integer board(1:8, 1:8)
integer imove(1:8, 1:64)
integer jmove(1:8, 1:64)
integer nmove(1:64)
integer n
board(imove(nmove(n), n), jmove(nmove(n), n)) = n
end
C-----------------------------------------------------------------------
subroutine unmove (board, imove, jmove, nmove, n)
implicit none
C Unmake a mkmove.
integer board(1:8, 1:8)
integer imove(1:8, 1:64)
integer jmove(1:8, 1:64)
integer nmove(1:64)
integer n
board(imove(nmove(n), n), jmove(nmove(n), n)) = -1
end
C-----------------------------------------------------------------------
function goodmv (imove, nmove, n)
implicit none
logical goodmv
integer imove(1:8, 1:64)
integer nmove(1:64)
integer n
goodmv = (imove(nmove(n), n) .ne. -1)
end
C-----------------------------------------------------------------------
subroutine prnt (board, itours)
implicit none
C Print a knight's tour.
integer board(1:8,1:8)
integer itours
10000 format (1X)
C The following plethora of format statements seemed a simple way to
C get this working with f2c. (For gfortran, the 'I0' format
C sufficed.)
10010 format (1X, "Tour number ", I1)
10020 format (1X, "Tour number ", I2)
10030 format (1X, "Tour number ", I3)
10040 format (1X, "Tour number ", I4)
10050 format (1X, "Tour number ", I5)
10060 format (1X, "Tour number ", I6)
10070 format (1X, "Tour number ", I20)
if (itours .lt. 10) then
write (*, 10010) itours
else if (itours .lt. 100) then
write (*, 10020) itours
else if (itours .lt. 1000) then
write (*, 10030) itours
else if (itours .lt. 10000) then
write (*, 10040) itours
else if (itours .lt. 100000) then
write (*, 10050) itours
else if (itours .lt. 1000000) then
write (*, 10060) itours
else
write (*, 10070) itours
end if
call prntmv (board)
call prntbd (board)
write (*, 10000)
end
C-----------------------------------------------------------------------
subroutine prntbd (board)
implicit none
C Print a chessboard with the move number in each square.
integer board(1:8,1:8)
integer i, j
10000 format (1X, " ", 8("+----"), "+")
10010 format (1X, I2, " ", 8(" | ", I2), " | ")
10020 format (1X, " ", 8(" ", A1))
do 1000 i = 8, 1, -1
write (*, 10000)
write (*, 10010) i, (board(i, j), j = 1, 8)
1000 continue
write (*, 10000)
write (*, 10020) 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'
end
C-----------------------------------------------------------------------
subroutine prntmv (board)
implicit none
C Print the moves of a knight's path, in algebraic notation.
integer board(1:8,1:8)
integer ipos(1:64)
integer jpos(1:64)
integer numpos
character*2 alg(1:64)
integer columns(1:8)
integer k
integer m
character*72 lines(1:8)
10000 format (1X, A)
call bd2pos (board, ipos, jpos, numpos)
C Convert the positions to algebraic notation.
do 1000 k = 1, numpos
call ij2alg (ipos(k), jpos(k), alg(k))
1000 continue
C Fill lines with algebraic notations.
do 1020 m = 1, 8
columns(m) = 1
1020 continue
m = 1
do 1100 k = 1, numpos
lines(m)(columns(m) : columns(m) + 1) = alg(k)(1:2)
columns(m) = columns(m) + 2
if (k .ne. numpos) then
lines(m)(columns(m) : columns(m) + 3) = " -> "
columns(m) = columns(m) + 4
else if (numpos .eq. 64 .and.
$ ((abs (ipos(numpos) - ipos(1)) .eq. 2
$ .and. abs (jpos(numpos) - jpos(1)) .eq. 1) .or.
$ ((abs (ipos(numpos) - ipos(1)) .eq. 1
$ .and. abs (jpos(numpos) - jpos(1)) .eq. 2)))) then
lines(m)(columns(m) : columns(m) + 8) = " -> cycle"
columns(m) = columns(m) + 9
endif
if (mod (k, 8) .eq. 0) m = m + 1
1100 continue
C Print the lines that have stuff in them.
do 1200 m = 1, 8
if (columns(m) .ne. 1) then
write (*, 10000) lines(m)(1 : columns(m) - 1)
end if
1200 continue
end
C-----------------------------------------------------------------------
function isclos (board)
implicit none
C Is a board a closed tour?
logical isclos
integer board(1:8,1:8)
integer ipos(1:64) ! The i-positions in order.
integer jpos(1:64) ! The j-positions in order.
integer numpos ! The number of positions so far.
call bd2pos (board, ipos, jpos, numpos)
isclos = (numpos .eq. 64 .and.
$ ((abs (ipos(numpos) - ipos(1)) .eq. 2
$ .and. abs (jpos(numpos) - jpos(1)) .eq. 1) .or.
$ ((abs (ipos(numpos) - ipos(1)) .eq. 1
$ .and. abs (jpos(numpos) - jpos(1)) .eq. 2))))
end
C-----------------------------------------------------------------------
subroutine bd2pos (board, ipos, jpos, numpos)
implicit none
C Convert from a board to a list of board positions.
integer board(1:8,1:8)
integer ipos(1:64) ! The i-positions in order.
integer jpos(1:64) ! The j-positions in order.
integer numpos ! The number of positions so far.
integer i, j
numpos = 0
do 1010 i = 1, 8
do 1000 j = 1, 8
if (board(i, j) .ne. -1) then
numpos = max (board(i, j), numpos)
ipos(board(i, j)) = i
jpos(board(i, j)) = j
end if
1000 continue
1010 continue
end
C-----------------------------------------------------------------------
subroutine nxtmov (board, n, imove, jmove, nmove)
implicit none
C Find possible next moves. Prune and sort the moves according to
C Warnsdorff's heuristic, keeping only those that have the minimum
C number of legal following moves.
integer board(1:8,1:8)
integer n
integer imove(1:8,1:64)
integer jmove(1:8,1:64)
integer nmove(1:64)
integer w1, w2, w3, w4, w5, w6, w7, w8
integer w
integer n1
integer pickw
call possib (board, n, imove, jmove, nmove)
n1 = n + 1
nmove(n1) = 1
call countf (board, n1, imove, jmove, nmove, w1)
nmove(n1) = 2
call countf (board, n1, imove, jmove, nmove, w2)
nmove(n1) = 3
call countf (board, n1, imove, jmove, nmove, w3)
nmove(n1) = 4
call countf (board, n1, imove, jmove, nmove, w4)
nmove(n1) = 5
call countf (board, n1, imove, jmove, nmove, w5)
nmove(n1) = 6
call countf (board, n1, imove, jmove, nmove, w6)
nmove(n1) = 7
call countf (board, n1, imove, jmove, nmove, w7)
nmove(n1) = 8
call countf (board, n1, imove, jmove, nmove, w8)
w = pickw (w1, w2, w3, w4, w5, w6, w7, w8)
if (w .eq. 0) then
call disabl (imove(1, n1), jmove(1, n1))
call disabl (imove(2, n1), jmove(2, n1))
call disabl (imove(3, n1), jmove(3, n1))
call disabl (imove(4, n1), jmove(4, n1))
call disabl (imove(5, n1), jmove(5, n1))
call disabl (imove(6, n1), jmove(6, n1))
call disabl (imove(7, n1), jmove(7, n1))
call disabl (imove(8, n1), jmove(8, n1))
else
if (w .ne. w1) call disabl (imove(1, n1), jmove(1, n1))
if (w .ne. w2) call disabl (imove(2, n1), jmove(2, n1))
if (w .ne. w3) call disabl (imove(3, n1), jmove(3, n1))
if (w .ne. w4) call disabl (imove(4, n1), jmove(4, n1))
if (w .ne. w5) call disabl (imove(5, n1), jmove(5, n1))
if (w .ne. w6) call disabl (imove(6, n1), jmove(6, n1))
if (w .ne. w7) call disabl (imove(7, n1), jmove(7, n1))
if (w .ne. w8) call disabl (imove(8, n1), jmove(8, n1))
end if
end
C-----------------------------------------------------------------------
subroutine countf (board, n, imove, jmove, nmove, w)
implicit none
C Count the number of moves possible after an nth move.
integer board(1:8,1:8)
integer n
integer imove(1:8,1:64)
integer jmove(1:8,1:64)
integer nmove(1:64)
integer w
logical goodmv
integer n1
if (goodmv (imove, nmove, n)) then
call mkmove (board, imove, jmove, nmove, n)
call possib (board, n, imove, jmove, nmove)
n1 = n + 1
w = 0
if (imove(1, n1) .ne. -1) w = w + 1
if (imove(2, n1) .ne. -1) w = w + 1
if (imove(3, n1) .ne. -1) w = w + 1
if (imove(4, n1) .ne. -1) w = w + 1
if (imove(5, n1) .ne. -1) w = w + 1
if (imove(6, n1) .ne. -1) w = w + 1
if (imove(7, n1) .ne. -1) w = w + 1
if (imove(8, n1) .ne. -1) w = w + 1
call unmove (board, imove, jmove, nmove, n)
else
C The nth move itself is impossible.
w = 0
end if
end
C-----------------------------------------------------------------------
function pickw (w1, w2, w3, w4, w5, w6, w7, w8)
implicit none
C From w1..w8, pick out the least nonzero value (or zero if they all
C equal zero).
integer pickw
integer w1, w2, w3, w4, w5, w6, w7, w8
integer w
integer pickw1
w = 0
w = pickw1 (w, w1)
w = pickw1 (w, w2)
w = pickw1 (w, w3)
w = pickw1 (w, w4)
w = pickw1 (w, w5)
w = pickw1 (w, w6)
w = pickw1 (w, w7)
w = pickw1 (w, w8)
pickw = w
end
C-----------------------------------------------------------------------
function pickw1 (u, v)
implicit none
C A small function used by pickw.
integer pickw1
integer u, v
if (v .eq. 0) then
pickw1 = u
else if (u .eq. 0) then
pickw1 = v
else
pickw1 = min (u, v)
end if
end
C-----------------------------------------------------------------------
subroutine possib (board, n, imove, jmove, nmove)
implicit none
C Find moves that are possible from an nth-move position.
integer board(1:8,1:8)
integer n
integer imove(1:8,1:64)
integer jmove(1:8,1:64)
integer nmove(1:64)
integer i, j
integer n1
i = imove(nmove(n), n)
j = jmove(nmove(n), n)
n1 = n + 1
call trymov (board, i + 1, j + 2, imove(1, n1), jmove(1, n1))
call trymov (board, i + 2, j + 1, imove(2, n1), jmove(2, n1))
call trymov (board, i + 1, j - 2, imove(3, n1), jmove(3, n1))
call trymov (board, i + 2, j - 1, imove(4, n1), jmove(4, n1))
call trymov (board, i - 1, j + 2, imove(5, n1), jmove(5, n1))
call trymov (board, i - 2, j + 1, imove(6, n1), jmove(6, n1))
call trymov (board, i - 1, j - 2, imove(7, n1), jmove(7, n1))
call trymov (board, i - 2, j - 1, imove(8, n1), jmove(8, n1))
end
C-----------------------------------------------------------------------
subroutine trymov (board, i, j, imove, jmove)
implicit none
C Try a move to square (i, j).
integer board(1:8,1:8)
integer i, j
integer imove, jmove
call disabl (imove, jmove)
if (1 .le. i .and. i .le. 8 .and. 1 .le. j .and. j .le. 8) then
if (board(i,j) .eq. -1) then
call enable (i, j, imove, jmove)
end if
end if
end
C-----------------------------------------------------------------------
subroutine enable (i, j, imove, jmove)
implicit none
C Enable a potential move.
integer i, j
integer imove, jmove
imove = i
jmove = j
end
C-----------------------------------------------------------------------
subroutine disabl (imove, jmove)
implicit none
C Disable a potential move.
integer imove, jmove
imove = -1
jmove = -1
end
C-----------------------------------------------------------------------
subroutine alg2ij (alg, i, j)
implicit none
C Convert, for instance, 'c5' to i=3,j=5.
character*2 alg
integer i, j
if (alg(1:1) .eq. 'a') j = 1
if (alg(1:1) .eq. 'b') j = 2
if (alg(1:1) .eq. 'c') j = 3
if (alg(1:1) .eq. 'd') j = 4
if (alg(1:1) .eq. 'e') j = 5
if (alg(1:1) .eq. 'f') j = 6
if (alg(1:1) .eq. 'g') j = 7
if (alg(1:1) .eq. 'h') j = 8
if (alg(2:2) .eq. '1') i = 1
if (alg(2:2) .eq. '2') i = 2
if (alg(2:2) .eq. '3') i = 3
if (alg(2:2) .eq. '4') i = 4
if (alg(2:2) .eq. '5') i = 5
if (alg(2:2) .eq. '6') i = 6
if (alg(2:2) .eq. '7') i = 7
if (alg(2:2) .eq. '8') i = 8
end
C-----------------------------------------------------------------------
subroutine ij2alg (i, j, alg)
implicit none
C Convert, for instance, i=3,j=5 to 'c5'.
integer i, j
character*2 alg
character alg1
character alg2
if (j .eq. 1) alg1 = 'a'
if (j .eq. 2) alg1 = 'b'
if (j .eq. 3) alg1 = 'c'
if (j .eq. 4) alg1 = 'd'
if (j .eq. 5) alg1 = 'e'
if (j .eq. 6) alg1 = 'f'
if (j .eq. 7) alg1 = 'g'
if (j .eq. 8) alg1 = 'h'
if (i .eq. 1) alg2 = '1'
if (i .eq. 2) alg2 = '2'
if (i .eq. 3) alg2 = '3'
if (i .eq. 4) alg2 = '4'
if (i .eq. 5) alg2 = '5'
if (i .eq. 6) alg2 = '6'
if (i .eq. 7) alg2 = '7'
if (i .eq. 8) alg2 = '8'
alg(1:1) = alg1
alg(2:2) = alg2
end
C-----------------------------------------------------------------------
- Output:
$ echo "c5 2 T" | ./knights_tour
Tour number 1
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> e8 -> d6 -> b5 -> d4 -> f5 -> g7 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8 | 56 | 3 | 50 | 21 | 58 | 5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7 | 51 | 22 | 57 | 4 | 49 | 20 | 63 | 6 |
+----+----+----+----+----+----+----+----+
6 | 2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5 | 23 | 60 | 1 | 48 | 53 | 62 | 7 | 46 |
+----+----+----+----+----+----+----+----+
4 | 38 | 13 | 54 | 61 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3 | 27 | 24 | 37 | 14 | 41 | 30 | 33 | 8 |
+----+----+----+----+----+----+----+----+
2 | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1 | 25 | 28 | 11 | 40 | 15 | 32 | 9 | 34 |
+----+----+----+----+----+----+----+----+
a b c d e f g h
Tour number 2
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> b5 -> d6 -> e8 -> g7 -> f5 -> d4 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8 | 56 | 3 | 50 | 21 | 60 | 5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7 | 51 | 22 | 57 | 4 | 49 | 20 | 61 | 6 |
+----+----+----+----+----+----+----+----+
6 | 2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5 | 23 | 58 | 1 | 48 | 53 | 62 | 7 | 46 |
+----+----+----+----+----+----+----+----+
4 | 38 | 13 | 54 | 63 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3 | 27 | 24 | 37 | 14 | 41 | 30 | 33 | 8 |
+----+----+----+----+----+----+----+----+
2 | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1 | 25 | 28 | 11 | 40 | 15 | 32 | 9 | 34 |
+----+----+----+----+----+----+----+----+
a b c d e f g h
Fortran 95
!-----------------------------------------------------------------------
!
! Find Knight’s Tours.
!
! Using Warnsdorff’s heuristic, find multiple solutions.
! Optionally accept only closed tours.
!
! This program is migrated from my implementation for
! ATS/Postiats. Unlike my FORTRAN 77 implementation (which simply
! cannot do so), it uses a recursive call.
!
! Compile with, for instance:
!
! gfortran -O2 -g -std=f95 -o knights_tour knights_tour.f90
!
! Usage examples:
!
! One tour starting at a1, either open or closed:
!
! echo "a1 1 F" | ./knights_tour
!
! No more than 2000 closed tours starting at c5:
!
! echo "c5 2000 T" | ./knights_tour
!
!-----------------------------------------------------------------------
program knights_tour
implicit none
character(len = 2) inp__alg
integer inp__istart
integer inp__jstart
integer inp__max_tours
logical inp__closed
read (*,*) inp__alg, inp__max_tours, inp__closed
call alg2ij (inp__alg, inp__istart, inp__jstart)
call main (inp__istart, inp__jstart, inp__max_tours, inp__closed)
contains
subroutine main (istart, jstart, max_tours, closed)
integer, intent(in) :: istart, jstart ! The starting position.
integer, intent(in) :: max_tours ! The max. no. of tours to print.
logical, intent(in) :: closed ! Closed tours only?
integer board(1:8,1:8)
integer num_tours_printed
num_tours_printed = 0
call init_board (board)
call explore (board, 1, istart, jstart, max_tours, &
& num_tours_printed, closed)
end subroutine main
recursive subroutine explore (board, n, i, j, max_tours, &
& num_tours_printed, closed)
! Recursively the space of 'Warnsdorffian' knight’s paths, looking
! for and printing complete tours.
integer, intent(inout) :: board(1:8,1:8)
integer, intent(in) :: n
integer, intent(in) :: i, j
integer, intent(in) :: max_tours
integer, intent(inout) :: num_tours_printed
logical, intent(in) :: closed
integer imove(1:8)
integer jmove(1:8)
integer k
if (num_tours_printed < max_tours .and. n /= 0) then
if (is_good_move (i, j)) then
call mkmove (board, i, j, n)
if (n == 63) then
call find_possible_moves (board, i, j, imove, jmove)
call try_last_move (board, n + 1, imove(1), jmove(1), &
& num_tours_printed, closed)
call try_last_move (board, n + 1, imove(2), jmove(2), &
& num_tours_printed, closed)
call try_last_move (board, n + 1, imove(3), jmove(3), &
& num_tours_printed, closed)
call try_last_move (board, n + 1, imove(4), jmove(4), &
& num_tours_printed, closed)
call try_last_move (board, n + 1, imove(5), jmove(5), &
& num_tours_printed, closed)
call try_last_move (board, n + 1, imove(6), jmove(6), &
& num_tours_printed, closed)
call try_last_move (board, n + 1, imove(7), jmove(7), &
& num_tours_printed, closed)
call try_last_move (board, n + 1, imove(8), jmove(8), &
& num_tours_printed, closed)
else
call find_next_moves (board, n, i, j, imove, jmove)
do k = 1, 8
if (is_good_move (imove(k), jmove(k))) then
!
! Here is the recursive call.
!
call explore (board, n + 1, imove(k), jmove(k), &
& max_tours, num_tours_printed, closed)
end if
end do
end if
call unmove (board, i, j)
end if
end if
end subroutine explore
subroutine try_last_move (board, n, i, j, num_tours_printed, closed)
integer, intent(inout) :: board(1:8,1:8)
integer, intent(in) :: n
integer, intent(in) :: i, j
integer, intent(inout) :: num_tours_printed
logical, intent(in) :: closed
integer ipos(1:64)
integer jpos(1:64)
integer numpos
integer idiff
integer jdiff
if (is_good_move (i, j)) then
call mkmove (board, i, j, n)
if (.not. closed) then
num_tours_printed = num_tours_printed + 1
call print_tour (board, num_tours_printed)
else
call board2positions (board, ipos, jpos, numpos)
idiff = abs (i - ipos(1))
jdiff = abs (j - jpos(1))
if ((idiff == 1 .and. jdiff == 2) .or. &
(idiff == 2 .and. jdiff == 1)) then
num_tours_printed = num_tours_printed + 1
call print_tour (board, num_tours_printed)
end if
end if
call unmove (board, i, j)
end if
end subroutine try_last_move
subroutine init_board (board)
! Initialize a chessboard with empty squares.
integer, intent(out) :: board(1:8,1:8)
integer i, j
do j = 1, 8
do i = 1, 8
board(i, j) = -1
end do
end do
end subroutine init_board
subroutine mkmove (board, i, j, n)
! Fill a square with a move number.
integer, intent(inout) :: board(1:8, 1:8)
integer, intent(in) :: i, j
integer, intent(in) :: n
board(i, j) = n
end subroutine mkmove
subroutine unmove (board, i, j)
! Unmake a mkmove.
integer, intent(inout) :: board(1:8, 1:8)
integer, intent(in) :: i, j
board(i, j) = -1
end subroutine unmove
function is_good_move (i, j)
logical is_good_move
integer, intent(in) :: i, j
is_good_move = (i /= -1 .and. j /= -1)
end function is_good_move
subroutine print_tour (board, num_tours_printed)
! Print a knight's tour.
integer, intent(in) :: board(1:8,1:8)
integer, intent(in) :: num_tours_printed
write (*, '("Tour number ", I0)') num_tours_printed
call print_moves (board)
call print_board (board)
write (*, '()')
end subroutine print_tour
subroutine print_board (board)
! Print a chessboard with the move number in each square.
integer, intent(in) :: board(1:8,1:8)
integer i, j
do i = 8, 1, -1
write (*, '(" ", 8("+----"), "+")')
write (*, '(I2, " ", 8(" | ", I2), " | ")') &
i, (board(i, j), j = 1, 8)
end do
write (*, '(" ", 8("+----"), "+")')
write (*, '(" ", 8(" ", A1))') &
'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'
end subroutine print_board
subroutine print_moves (board)
! Print the moves of a knight's path, in algebraic notation.
integer, intent(in) :: board(1:8,1:8)
integer ipos(1:64)
integer jpos(1:64)
integer numpos
character(len = 2) alg(1:64)
integer columns(1:8)
integer k
integer m
character(len = 72) lines(1:8)
call board2positions (board, ipos, jpos, numpos)
! Convert the positions to algebraic notation.
do k = 1, numpos
call ij2alg (ipos(k), jpos(k), alg(k))
end do
! Fill lines with algebraic notations.
do m = 1, 8
columns(m) = 1
end do
m = 1
do k = 1, numpos
lines(m)(columns(m) : columns(m) + 1) = alg(k)(1:2)
columns(m) = columns(m) + 2
if (k /= numpos) then
lines(m)(columns(m) : columns(m) + 3) = " -> "
columns(m) = columns(m) + 4
else if (numpos == 64 .and. &
((abs (ipos(numpos) - ipos(1)) == 2 &
.and. abs (jpos(numpos) - jpos(1)) == 1) .or. &
((abs (ipos(numpos) - ipos(1)) == 1 &
.and. abs (jpos(numpos) - jpos(1)) == 2)))) then
lines(m)(columns(m) : columns(m) + 8) = " -> cycle"
columns(m) = columns(m) + 9
endif
if (mod (k, 8) == 0) m = m + 1
end do
! Print the lines that have stuff in them.
do m = 1, 8
if (columns(m) /= 1) then
write (*, '(A)') lines(m)(1 : columns(m) - 1)
end if
end do
end subroutine print_moves
function is_closed (board)
! Is a board a closed tour?
logical is_closed
integer board(1:8,1:8)
integer ipos(1:64) ! The i-positions in order.
integer jpos(1:64) ! The j-positions in order.
integer numpos ! The number of positions so far.
call board2positions (board, ipos, jpos, numpos)
is_closed = (numpos == 64 .and. &
((abs (ipos(numpos) - ipos(1)) == 2 &
.and. abs (jpos(numpos) - jpos(1)) == 1) .or. &
((abs (ipos(numpos) - ipos(1)) == 1 &
.and. abs (jpos(numpos) - jpos(1)) == 2))))
end function is_closed
subroutine board2positions (board, ipos, jpos, numpos)
! Convert from a board to a list of board positions.
integer, intent(in) :: board(1:8,1:8)
integer, intent(out) :: ipos(1:64) ! The i-positions in order.
integer, intent(out) :: jpos(1:64) ! The j-positions in order.
integer, intent(out) :: numpos ! The number of positions so far.
integer i, j
numpos = 0
do i = 1, 8
do j = 1, 8
if (board(i, j) /= -1) then
numpos = max (board(i, j), numpos)
ipos(board(i, j)) = i
jpos(board(i, j)) = j
end if
end do
end do
end subroutine board2positions
subroutine find_next_moves (board, n, i, j, imove, jmove)
! Find possible next moves. Prune and sort the moves according to
! Warnsdorff's heuristic, keeping only those that have the minimum
! number of legal following moves.
integer, intent(inout) :: board(1:8,1:8)
integer, intent(in) :: n
integer, intent(in) :: i, j
integer, intent(inout) :: imove(1:8)
integer, intent(inout) :: jmove(1:8)
integer w1, w2, w3, w4, w5, w6, w7, w8
integer w
call find_possible_moves (board, i, j, imove, jmove)
call count_following (board, n + 1, imove(1), jmove(1), w1)
call count_following (board, n + 1, imove(2), jmove(2), w2)
call count_following (board, n + 1, imove(3), jmove(3), w3)
call count_following (board, n + 1, imove(4), jmove(4), w4)
call count_following (board, n + 1, imove(5), jmove(5), w5)
call count_following (board, n + 1, imove(6), jmove(6), w6)
call count_following (board, n + 1, imove(7), jmove(7), w7)
call count_following (board, n + 1, imove(8), jmove(8), w8)
w = pick_w (w1, w2, w3, w4, w5, w6, w7, w8)
if (w == 0) then
call disable (imove(1), jmove(1))
call disable (imove(2), jmove(2))
call disable (imove(3), jmove(3))
call disable (imove(4), jmove(4))
call disable (imove(5), jmove(5))
call disable (imove(6), jmove(6))
call disable (imove(7), jmove(7))
call disable (imove(8), jmove(8))
else
if (w /= w1) call disable (imove(1), jmove(1))
if (w /= w2) call disable (imove(2), jmove(2))
if (w /= w3) call disable (imove(3), jmove(3))
if (w /= w4) call disable (imove(4), jmove(4))
if (w /= w5) call disable (imove(5), jmove(5))
if (w /= w6) call disable (imove(6), jmove(6))
if (w /= w7) call disable (imove(7), jmove(7))
if (w /= w8) call disable (imove(8), jmove(8))
end if
end subroutine find_next_moves
subroutine count_following (board, n, i, j, w)
! Count the number of moves possible after an nth move.
integer, intent(inout) :: board(1:8,1:8)
integer, intent(in) :: n
integer, intent(in) :: i, j
integer, intent(out) :: w
integer imove(1:8)
integer jmove(1:8)
if (is_good_move (i, j)) then
call mkmove (board, i, j, n)
call find_possible_moves (board, i, j, imove, jmove)
w = 0
if (is_good_move (imove(1), jmove(1))) w = w + 1
if (is_good_move (imove(2), jmove(2))) w = w + 1
if (is_good_move (imove(3), jmove(3))) w = w + 1
if (is_good_move (imove(4), jmove(4))) w = w + 1
if (is_good_move (imove(5), jmove(5))) w = w + 1
if (is_good_move (imove(6), jmove(6))) w = w + 1
if (is_good_move (imove(7), jmove(7))) w = w + 1
if (is_good_move (imove(8), jmove(8))) w = w + 1
call unmove (board, i, j)
else
! The nth move itself is impossible.
w = 0
end if
end subroutine count_following
function pick_w (w1, w2, w3, w4, w5, w6, w7, w8) result (w)
! From w1..w8, pick out the least nonzero value (or zero if they
! all equal zero).
integer, intent(in) :: w1, w2, w3, w4, w5, w6, w7, w8
integer w
w = 0
w = pick_w1 (w, w1)
w = pick_w1 (w, w2)
w = pick_w1 (w, w3)
w = pick_w1 (w, w4)
w = pick_w1 (w, w5)
w = pick_w1 (w, w6)
w = pick_w1 (w, w7)
w = pick_w1 (w, w8)
end function pick_w
function pick_w1 (u, v)
! A small function used by pick_w.
integer pick_w1
integer, intent(in) :: u, v
if (v == 0) then
pick_w1 = u
else if (u == 0) then
pick_w1 = v
else
pick_w1 = min (u, v)
end if
end function pick_w1
subroutine find_possible_moves (board, i, j, imove, jmove)
! Find moves that are possible from a position.
integer, intent(in) :: board(1:8,1:8)
integer, intent(in) :: i, j
integer, intent(out) :: imove(1:8)
integer, intent(out) :: jmove(1:8)
call trymov (board, i + 1, j + 2, imove(1), jmove(1))
call trymov (board, i + 2, j + 1, imove(2), jmove(2))
call trymov (board, i + 1, j - 2, imove(3), jmove(3))
call trymov (board, i + 2, j - 1, imove(4), jmove(4))
call trymov (board, i - 1, j + 2, imove(5), jmove(5))
call trymov (board, i - 2, j + 1, imove(6), jmove(6))
call trymov (board, i - 1, j - 2, imove(7), jmove(7))
call trymov (board, i - 2, j - 1, imove(8), jmove(8))
end subroutine find_possible_moves
subroutine trymov (board, i, j, imove, jmove)
! Try a move to square (i, j).
integer, intent(in) :: board(1:8,1:8)
integer, intent(in) :: i, j
integer, intent(inout) :: imove, jmove
call disable (imove, jmove)
if (1 <= i .and. i <= 8 .and. 1 <= j .and. j <= 8) then
if (square_is_empty (board, i, j)) then
call enable (i, j, imove, jmove)
end if
end if
end subroutine trymov
function square_is_empty (board, i, j)
logical square_is_empty
integer, intent(in) :: board(1:8,1:8)
integer, intent(in) :: i, j
square_is_empty = (board(i, j) == -1)
end function square_is_empty
subroutine enable (i, j, imove, jmove)
! Enable a potential move.
integer, intent(in) :: i, j
integer, intent(inout) :: imove, jmove
imove = i
jmove = j
end subroutine enable
subroutine disable (imove, jmove)
! Disable a potential move.
integer, intent(out) :: imove, jmove
imove = -1
jmove = -1
end subroutine disable
subroutine alg2ij (alg, i, j)
! Convert, for instance, 'c5' to i=3,j=5.
character(len = 2), intent(in) :: alg
integer, intent(out) :: i, j
if (alg(1:1) == 'a') j = 1
if (alg(1:1) == 'b') j = 2
if (alg(1:1) == 'c') j = 3
if (alg(1:1) == 'd') j = 4
if (alg(1:1) == 'e') j = 5
if (alg(1:1) == 'f') j = 6
if (alg(1:1) == 'g') j = 7
if (alg(1:1) == 'h') j = 8
if (alg(2:2) == '1') i = 1
if (alg(2:2) == '2') i = 2
if (alg(2:2) == '3') i = 3
if (alg(2:2) == '4') i = 4
if (alg(2:2) == '5') i = 5
if (alg(2:2) == '6') i = 6
if (alg(2:2) == '7') i = 7
if (alg(2:2) == '8') i = 8
end subroutine alg2ij
subroutine ij2alg (i, j, alg)
! Convert, for instance, i=3,j=5 to 'c5'.
integer, intent(in) :: i, j
character(len = 2), intent(out) :: alg
character alg1
character alg2
if (j == 1) alg1 = 'a'
if (j == 2) alg1 = 'b'
if (j == 3) alg1 = 'c'
if (j == 4) alg1 = 'd'
if (j == 5) alg1 = 'e'
if (j == 6) alg1 = 'f'
if (j == 7) alg1 = 'g'
if (j == 8) alg1 = 'h'
if (i == 1) alg2 = '1'
if (i == 2) alg2 = '2'
if (i == 3) alg2 = '3'
if (i == 4) alg2 = '4'
if (i == 5) alg2 = '5'
if (i == 6) alg2 = '6'
if (i == 7) alg2 = '7'
if (i == 8) alg2 = '8'
alg(1:1) = alg1
alg(2:2) = alg2
end subroutine ij2alg
end program
!-----------------------------------------------------------------------
- Output:
$ echo "c5 2 T" | ./knights_tour
Tour number 1
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> e8 -> d6 -> b5 -> d4 -> f5 -> g7 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8 | 56 | 3 | 50 | 21 | 58 | 5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7 | 51 | 22 | 57 | 4 | 49 | 20 | 63 | 6 |
+----+----+----+----+----+----+----+----+
6 | 2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5 | 23 | 60 | 1 | 48 | 53 | 62 | 7 | 46 |
+----+----+----+----+----+----+----+----+
4 | 38 | 13 | 54 | 61 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3 | 27 | 24 | 37 | 14 | 41 | 30 | 33 | 8 |
+----+----+----+----+----+----+----+----+
2 | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1 | 25 | 28 | 11 | 40 | 15 | 32 | 9 | 34 |
+----+----+----+----+----+----+----+----+
a b c d e f g h
Tour number 2
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> b5 -> d6 -> e8 -> g7 -> f5 -> d4 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8 | 56 | 3 | 50 | 21 | 60 | 5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7 | 51 | 22 | 57 | 4 | 49 | 20 | 61 | 6 |
+----+----+----+----+----+----+----+----+
6 | 2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5 | 23 | 58 | 1 | 48 | 53 | 62 | 7 | 46 |
+----+----+----+----+----+----+----+----+
4 | 38 | 13 | 54 | 63 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3 | 27 | 24 | 37 | 14 | 41 | 30 | 33 | 8 |
+----+----+----+----+----+----+----+----+
2 | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1 | 25 | 28 | 11 | 40 | 15 | 32 | 9 | 34 |
+----+----+----+----+----+----+----+----+
a b c d e f g h
Fortran 2008
(This one is not a translation of my ATS implementation. I wrote it earlier.)
!!!
!!! Find a Knight’s Tour.
!!!
!!! Use Warnsdorff’s heuristic, but write the program so it should not
!!! be able to terminate unsuccessfully.
!!!
module knights_tour
use, intrinsic :: iso_fortran_env, only: output_unit, error_unit
implicit none
private
public :: find_a_knights_tour
public :: notation_is_a_square
integer, parameter :: number_of_ranks = 8
integer, parameter :: number_of_files = 8
integer, parameter :: number_of_squares = number_of_ranks * number_of_files
! ‘Algebraic’ chess notation.
character, parameter :: rank_notation(1:8) = (/ '1', '2', '3', '4', '5', '6', '7', '8' /)
character, parameter :: file_notation(1:8) = (/ 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h' /)
type :: board_square_t
! Squares are represented by their algebraic notation.
character(2) :: algebraic_notation
contains
procedure, pass :: output => board_square_t_output
procedure, pass :: knight_moves => board_square_t_knight_moves
procedure, pass :: equal => board_square_t_equal
generic :: operator(==) => equal
end type board_square_t
type :: knight_moves_t
integer :: number_of_squares
type(board_square_t) :: squares(1:8)
end type knight_moves_t
type :: path_t
integer :: length
type(board_square_t) :: squares(1:number_of_squares)
contains
procedure, pass :: output => path_t_output
end type path_t
contains
pure function notation_is_a_square (notation) result (bool)
character(*), intent(in) :: notation
logical :: bool
integer :: length
integer :: rank_no
integer :: file_no
length = len_trim (notation)
if (length /= 2) then
bool = .false.
else
rank_no = findloc (rank_notation, notation(2:2), 1)
file_no = findloc (file_notation, notation(1:1), 1)
bool = (1 <= rank_no .and. rank_no <= number_of_ranks) &
& .and. (1 <= file_no .and. file_no <= number_of_files)
end if
end function notation_is_a_square
subroutine path_t_output (path, unit)
!
! Print a path in algebraic notation.
!
class(path_t), intent(in) :: path
integer, intent(in) :: unit
integer :: moves_counter
integer :: i
moves_counter = 1
if (1 <= path%length) then
call path%squares(1)%output(unit)
do i = 2, path%length
if (moves_counter == 8) then
write (unit, '(" ->")', advance = 'yes')
moves_counter = 1
else
write (unit, '(" -> ")', advance = 'no')
moves_counter = moves_counter + 1
end if
call path%squares(i)%output(unit)
end do
end if
write (output_unit, '()')
end subroutine path_t_output
subroutine board_square_t_output (square, unit)
!
! Print a square in algebraic notation.
!
class(board_square_t), intent(in) :: square
integer, intent(in) :: unit
write (unit, '(A2)', advance = 'no') square%algebraic_notation
end subroutine board_square_t_output
elemental function board_square_t_equal (p, q) result (bool)
class(board_square_t), intent(in) :: p, q
logical :: bool
bool = (p%algebraic_notation == q%algebraic_notation)
end function board_square_t_equal
pure function board_square_t_knight_moves (square) result (moves)
!
! Return all possible moves of a knight from a given square.
!
class(board_square_t), intent(in) :: square
type(knight_moves_t) :: moves
integer, parameter :: rank_stride(1:number_of_ranks) = (/ +1, +2, +1, +2, -1, -2, -1, -2 /)
integer, parameter :: file_stride(1:number_of_files) = (/ +2, +1, -2, -1, +2, +1, -2, -1 /)
integer :: rank_no, file_no
integer :: new_rank_no, new_file_no
integer :: i
character(2) :: notation
rank_no = findloc (rank_notation, square%algebraic_notation(2:2), 1)
file_no = findloc (file_notation, square%algebraic_notation(1:1), 1)
moves%number_of_squares = 0
do i = 1, 8
new_rank_no = rank_no + rank_stride(i)
new_file_no = file_no + file_stride(i)
if (1 <= new_rank_no &
& .and. new_rank_no <= number_of_ranks &
& .and. 1 <= new_file_no &
& .and. new_file_no <= number_of_files) then
moves%number_of_squares = moves%number_of_squares + 1
notation(2:2) = rank_notation(new_rank_no)
notation(1:1) = file_notation(new_file_no)
moves%squares(moves%number_of_squares) = board_square_t (notation)
end if
end do
end function board_square_t_knight_moves
pure function unvisited_knight_moves (path) result (moves)
!
! Return moves of a knight from a given square, but only those
! that have not been visited already.
!
class(path_t), intent(in) :: path
type(knight_moves_t) :: moves
type(knight_moves_t) :: all_moves
integer :: i
all_moves = path%squares(path%length)%knight_moves()
moves%number_of_squares = 0
do i = 1, all_moves%number_of_squares
if (all (.not. all_moves%squares(i) == path%squares(1:path%length))) then
moves%number_of_squares = moves%number_of_squares + 1
moves%squares(moves%number_of_squares) = all_moves%squares(i)
end if
end do
end function unvisited_knight_moves
pure function potential_knight_moves (path) result (moves)
!
! Return moves of a knight from a given square, but only those
! that are unvisited, and from which another unvisited move can be
! made.
!
! Sort the returned moves in nondecreasing order of the number of
! possible moves after the first. (This is how we implement
! Warnsdorff’s heuristic.)
!
class(path_t), intent(in) :: path
type(knight_moves_t) :: moves
type(knight_moves_t) :: unvisited_moves
type(knight_moves_t) :: next_moves
type(path_t) :: next_path
type(board_square_t) :: unpruned_squares(1:8)
integer :: warnsdorff_numbers(1:8)
integer :: number_of_unpruned_squares
integer :: i
if (path%length == number_of_squares - 1) then
!
! There is only one square left on the board. Either the knight
! can reach it or it cannot.
!
moves = unvisited_knight_moves (path)
else
!
! Use Warnsdorff’s heuristic: return unvisited moves, but try
! first those with the least number of possible moves following
! it.
!
! If the number of possible moves following is zero, prune the
! move, because it is a dead end.
!
number_of_unpruned_squares = 0
unvisited_moves = unvisited_knight_moves (path)
do i = 1, unvisited_moves%number_of_squares
next_path%length = path%length + 1
next_path%squares(1:path%length) = path%squares(1:path%length)
next_path%squares(next_path%length) = unvisited_moves%squares(i)
next_moves = unvisited_knight_moves (next_path)
if (next_moves%number_of_squares /= 0) then
number_of_unpruned_squares = number_of_unpruned_squares + 1
unpruned_squares(number_of_unpruned_squares) = unvisited_moves%squares(i)
warnsdorff_numbers(number_of_unpruned_squares) = next_moves%number_of_squares
end if
end do
! In-place insertion sort of the unpruned squares.
block
type(board_square_t) :: square
integer :: w_number
integer :: i, j
i = 2
do while (i <= number_of_unpruned_squares)
square = unpruned_squares(i)
w_number = warnsdorff_numbers(i)
j = i - 1
do while (1 <= j .and. w_number < warnsdorff_numbers(j))
unpruned_squares(j + 1) = unpruned_squares(j)
warnsdorff_numbers(j + 1) = warnsdorff_numbers(j)
j = j - 1
end do
unpruned_squares(j + 1) = square
warnsdorff_numbers(j + 1) = w_number
i = i + 1
end do
end block
moves%number_of_squares = number_of_unpruned_squares
moves%squares(1:number_of_unpruned_squares) = &
& unpruned_squares(1:number_of_unpruned_squares)
end if
end function potential_knight_moves
subroutine find_a_knights_tour (starting_square)
!
! Find and print a full knight’s tour.
!
character(2), intent(in) :: starting_square
type(path_t) :: path
path%length = 1
path%squares(1) = board_square_t (starting_square)
path = try_paths (path)
if (path%length /= 0) then
call path%output(output_unit)
else
write (error_unit, '("The program terminated without finding a solution.")')
write (error_unit, '("This is supposed to be impossible for an 8-by-8 board.")')
write (error_unit, '("The program is wrong.")')
error stop
end if
contains
recursive function try_paths (path) result (solution)
!
! Recursively try all possible paths, but using Warnsdorff’s
! heuristic to speed up the search.
!
class(path_t), intent(in) :: path
type(path_t) :: solution
type(path_t) :: new_path
type(knight_moves_t) :: moves
integer :: i
if (path%length == number_of_squares) then
solution = path
else
solution%length = 0
moves = potential_knight_moves (path)
if (moves%number_of_squares /= 0) then
new_path%length = path%length + 1
new_path%squares(1:path%length) = path%squares(1:path%length)
i = 1
do while (solution%length == 0 .and. i <= moves%number_of_squares)
new_path%squares(new_path%length) = moves%squares(i)
solution = try_paths (new_path)
i = i + 1
end do
end if
end if
end function try_paths
end subroutine find_a_knights_tour
end module knights_tour
program knights_tour_main
use, intrinsic :: iso_fortran_env, only: output_unit
use, non_intrinsic :: knights_tour
implicit none
character(200) :: arg
integer :: arg_count
integer :: i
arg_count = command_argument_count ()
do i = 1, arg_count
call get_command_argument (i, arg)
arg = adjustl (arg)
if (1 < i) write (output_unit, '()')
if (notation_is_a_square (arg)) then
call find_a_knights_tour (arg)
else
write (output_unit, '("This is not algebraic notation: ", A)') arg
end if
end do
end program knights_tour_main
$ ./knights_tour a1 b2 c3
a1 -> c2 -> a3 -> b1 -> d2 -> f1 -> h2 -> g4 -> h6 -> g8 -> e7 -> c8 -> a7 -> b5 -> c7 -> a8 -> b6 -> a4 -> b2 -> d1 -> f2 -> h1 -> g3 -> h5 -> g7 -> e8 -> f6 -> h7 -> f8 -> d7 -> b8 -> a6 -> b4 -> a2 -> c3 -> d5 -> e3 -> f5 -> h4 -> g2 -> e1 -> f3 -> g1 -> h3 -> g5 -> e4 -> d6 -> c4 -> a5 -> b7 -> d8 -> f7 -> h8 -> g6 -> e5 -> c6 -> d4 -> e6 -> f4 -> e2 -> c1 -> d3 -> c5 -> b3 b2 -> a4 -> b6 -> a8 -> c7 -> e8 -> g7 -> h5 -> g3 -> h1 -> f2 -> d1 -> c3 -> a2 -> c1 -> e2 -> g1 -> h3 -> f4 -> g2 -> h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 -> a1 -> c2 -> e1 -> d3 -> b4 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> e6 -> c5 -> e4 -> f6 -> g8 -> h6 -> g4 -> h2 -> f1 -> d2 -> b1 -> a3 -> c4 -> e5 -> f3 -> d4 -> b5 -> d6 -> c8 -> a7 -> c6 -> e7 -> f5 -> e3 -> d5 c3 -> a2 -> c1 -> e2 -> g1 -> h3 -> g5 -> h7 -> f8 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 -> a1 -> c2 -> a3 -> b1 -> d2 -> f1 -> h2 -> f3 -> h4 -> g2 -> e1 -> d3 -> f4 -> h5 -> g7 -> e8 -> f6 -> g8 -> h6 -> g4 -> e5 -> d7 -> b8 -> a6 -> b4 -> c6 -> a7 -> c8 -> e7 -> d5 -> b6 -> a8 -> c7 -> e6 -> c5 -> a4 -> b2 -> c4 -> e3 -> d1 -> f2 -> h1 -> g3 -> e4 -> d6 -> f5 -> d4 -> b5
Go
Warnsdorf's rule
package main
import (
"fmt"
"math/rand"
"time"
)
// input, 0-based start position
const startRow = 0
const startCol = 0
func main() {
rand.Seed(time.Now().Unix())
for !knightTour() {
}
}
var moves = []struct{ dr, dc int }{
{2, 1},
{2, -1},
{1, 2},
{1, -2},
{-1, 2},
{-1, -2},
{-2, 1},
{-2, -1},
}
// Attempt knight tour starting at startRow, startCol using Warnsdorff's rule
// and random tie breaking. If a tour is found, print it and return true.
// Otherwise no backtracking, just return false.
func knightTour() bool {
// 8x8 board. squares hold 1-based visit order. 0 means unvisited.
board := make([][]int, 8)
for i := range board {
board[i] = make([]int, 8)
}
r := startRow
c := startCol
board[r][c] = 1 // first move
for move := 2; move <= 64; move++ {
minNext := 8
var mr, mc, nm int
candidateMoves:
for _, cm := range moves {
cr := r + cm.dr
if cr < 0 || cr >= 8 { // off board
continue
}
cc := c + cm.dc
if cc < 0 || cc >= 8 { // off board
continue
}
if board[cr][cc] > 0 { // already visited
continue
}
// cr, cc candidate legal move.
p := 0 // count possible next moves.
for _, m2 := range moves {
r2 := cr + m2.dr
if r2 < 0 || r2 >= 8 {
continue
}
c2 := cc + m2.dc
if c2 < 0 || c2 >= 8 {
continue
}
if board[r2][c2] > 0 {
continue
}
p++
if p > minNext { // bail out as soon as it's eliminated
continue candidateMoves
}
}
if p < minNext { // it's better. keep it.
minNext = p // new min possible next moves
nm = 1 // number of candidates with this p
mr = cr // best candidate move
mc = cc
continue
}
// it ties for best so far.
// keep it with probability 1/(number of tying moves)
nm++ // number of tying moves
if rand.Intn(nm) == 0 { // one chance to keep it
mr = cr
mc = cc
}
}
if nm == 0 { // no legal move
return false
}
// make selected move
r = mr
c = mc
board[r][c] = move
}
// tour complete. print board.
for _, r := range board {
for _, m := range r {
fmt.Printf("%3d", m)
}
fmt.Println()
}
return true
}
- Output:
1 4 39 20 23 6 63 58 40 19 2 5 62 57 22 7 3 38 41 48 21 24 59 64 18 43 32 37 56 61 8 25 31 14 47 42 49 36 53 60 46 17 44 33 52 55 26 9 13 30 15 50 11 28 35 54 16 45 12 29 34 51 10 27
Ant colony
/* Adapted from "Enumerating Knight's Tours using an Ant Colony Algorithm"
by Philip Hingston and Graham Kendal,
PDF at http://www.cs.nott.ac.uk/~gxk/papers/cec05knights.pdf. */
package main
import (
"fmt"
"math/rand"
"sync"
"time"
)
const boardSize = 8
const nSquares = boardSize * boardSize
const completeTour = nSquares - 1
// task input: starting square. These are 1 based, but otherwise 0 based
// row and column numbers are used througout the program.
const rStart = 2
const cStart = 3
// pheromone representation read by ants
var tNet = make([]float64, nSquares*8)
// row, col deltas of legal moves
var drc = [][]int{{1, 2}, {2, 1}, {2, -1}, {1, -2},
{-1, -2}, {-2, -1}, {-2, 1}, {-1, 2}}
// get square reached by following edge k from square (r, c)
func dest(r, c, k int) (int, int, bool) {
r += drc[k][0]
c += drc[k][1]
return r, c, r >= 0 && r < boardSize && c >= 0 && c < boardSize
}
// struct represents a pheromone amount associated with a move
type rckt struct {
r, c, k int
t float64
}
func main() {
fmt.Println("Starting square: row", rStart, "column", cStart)
// initialize board
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
for k := 0; k < 8; k++ {
if _, _, ok := dest(r, c, k); ok {
tNet[(r*boardSize+c)*8+k] = 1e-6
}
}
}
}
// waitGroups for ant release clockwork
var start, reset sync.WaitGroup
start.Add(1)
// channel for ants to return tours with pheremone updates
tch := make(chan []rckt)
// create an ant for each square
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
go ant(r, c, &start, &reset, tch)
}
}
// accumulator for new pheromone amounts
tNew := make([]float64, nSquares*8)
// each iteration is a "cycle" as described in the paper
for {
// evaporate pheromones
for i := range tNet {
tNet[i] *= .75
}
reset.Add(nSquares) // number of ants to release
start.Done() // release them
reset.Wait() // wait for them to begin searching
start.Add(1) // reset start signal for next cycle
// gather tours from ants
for i := 0; i < nSquares; i++ {
tour := <-tch
// watch for a complete tour from the specified starting square
if len(tour) == completeTour &&
tour[0].r == rStart-1 && tour[0].c == cStart-1 {
// task output: move sequence in a grid.
seq := make([]int, nSquares)
for i, sq := range tour {
seq[sq.r*boardSize+sq.c] = i + 1
}
last := tour[len(tour)-1]
r, c, _ := dest(last.r, last.c, last.k)
seq[r*boardSize+c] = nSquares
fmt.Println("Move sequence:")
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
fmt.Printf(" %3d", seq[r*boardSize+c])
}
fmt.Println()
}
return // task only requires finding a single tour
}
// accumulate pheromone amounts from all ants
for _, move := range tour {
tNew[(move.r*boardSize+move.c)*8+move.k] += move.t
}
}
// update pheromone amounts on network, reset accumulator
for i, tn := range tNew {
tNet[i] += tn
tNew[i] = 0
}
}
}
type square struct {
r, c int
}
func ant(r, c int, start, reset *sync.WaitGroup, tourCh chan []rckt) {
rnd := rand.New(rand.NewSource(time.Now().UnixNano()))
tabu := make([]square, nSquares)
moves := make([]rckt, nSquares)
unexp := make([]rckt, 8)
tabu[0].r = r
tabu[0].c = c
for {
// cycle initialization
moves = moves[:0]
tabu = tabu[:1]
r := tabu[0].r
c := tabu[0].c
// wait for start signal
start.Wait()
reset.Done()
for {
// choose next move
unexp = unexp[:0]
var tSum float64
findU:
for k := 0; k < 8; k++ {
dr, dc, ok := dest(r, c, k)
if !ok {
continue
}
for _, t := range tabu {
if t.r == dr && t.c == dc {
continue findU
}
}
tk := tNet[(r*boardSize+c)*8+k]
tSum += tk
// note: dest r, c stored here
unexp = append(unexp, rckt{dr, dc, k, tk})
}
if len(unexp) == 0 {
break // no moves
}
rn := rnd.Float64() * tSum
var move rckt
for _, move = range unexp {
if rn <= move.t {
break
}
rn -= move.t
}
// move to new square
move.r, r = r, move.r
move.c, c = c, move.c
tabu = append(tabu, square{r, c})
moves = append(moves, move)
}
// compute pheromone amount to leave
for i := range moves {
moves[i].t = float64(len(moves)-i) / float64(completeTour-i)
}
// return tour found for this cycle
tourCh <- moves
}
}
Output:
Starting square: row 2 column 3 Move sequence: 64 33 36 3 54 49 38 51 35 4 1 30 37 52 55 48 32 63 34 53 2 47 50 39 5 18 31 46 29 20 13 56 62 27 44 19 14 11 40 21 17 6 15 28 45 22 57 12 26 61 8 43 24 59 10 41 7 16 25 60 9 42 23 58
Haskell
import Data.Bifunctor (bimap)
import Data.Char (chr, ord)
import Data.List (intercalate, minimumBy, sort, (\\))
import Data.Ord (comparing)
import Control.Monad (join)
---------------------- KNIGHT'S TOUR ---------------------
type Square = (Int, Int)
knightTour :: [Square] -> [Square]
knightTour moves
| null possibilities = reverse moves
| otherwise = knightTour $ newSquare : moves
where
newSquare =
minimumBy
(comparing (length . findMoves))
possibilities
possibilities = findMoves $ head moves
findMoves = (\\ moves) . knightOptions
knightOptions :: Square -> [Square]
knightOptions (x, y) =
knightMoves >>= go . bimap (+ x) (+ y)
where
go move
| uncurry (&&) (both onBoard move) = [move]
| otherwise = []
knightMoves :: [(Int, Int)]
knightMoves =
((>>=) <*> (\deltas n -> deltas >>= go n)) [1, 2, -1, -2]
where
go i x
| abs i /= abs x = [(i, x)]
| otherwise = []
onBoard :: Int -> Bool
onBoard = (&&) . (0 <) <*> (9 >)
both :: (a -> b) -> (a, a) -> (b, b)
both = join bimap
--------------------------- TEST -------------------------
startPoint :: String
startPoint = "e5"
algebraic :: (Int, Int) -> String
algebraic (x, y) = [chr (x + 96), chr (y + 48)]
main :: IO ()
main =
printTour $
algebraic
<$> knightTour
[(\[x, y] -> (ord x - 96, ord y - 48)) startPoint]
where
printTour [] = return ()
printTour tour = do
putStrLn $ intercalate " -> " $ take 8 tour
printTour $ drop 8 tour
- Output:
e5 -> f7 -> h8 -> g6 -> h4 -> g2 -> e1 -> f3 g1 -> h3 -> g5 -> h7 -> f8 -> d7 -> b8 -> a6 b4 -> a2 -> c1 -> d3 -> b2 -> d1 -> f2 -> h1 g3 -> h5 -> g7 -> e8 -> f6 -> g8 -> h6 -> g4 h2 -> f1 -> e3 -> f5 -> e7 -> c8 -> a7 -> c6 d8 -> b7 -> a5 -> b3 -> a1 -> c2 -> d4 -> e2 f4 -> e6 -> c5 -> a4 -> b6 -> a8 -> c7 -> d5 c3 -> e4 -> d6 -> b5 -> a3 -> b1 -> d2 -> c4
Icon and Unicon
This implements Warnsdorff's algorithm using unordered sets.
- The board must be square (it has only been tested on 8x8 and 7x7). Currently the maximum size board (due to square notation) is 26x26.
- Tie breaking is selectable with 3 variants supplied (first in list, random, and Roth's distance heuristic).
- A debug log can be generated showing the moves and choices considered for tie breaking.
The algorithm doesn't always generate a complete tour.